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Combinatorics of Acyclic Orientations of Graphs: Algebra, Geometry and Probability by

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Benjamin Iriarte Giraldo M.A., San Francisco State University (2010) B.S., Universidad de los Andes (2009)

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Submitted to the Department of Mathematics in partial fulfillment of the requirements for the degree of Doctor of Philosophy in Mathematics at the MASSACHUSETTS INSTITUTE OF TECHNOLOGY June 2015

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Department of Mathematics February 25, 2015 Certified by..

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Richard P. Stanley Professor of Applied Mathematics Thesis Supervisor

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lAfichel X. Goemans Chairman, Department Committee on Graduate Theses

Combinatorics of Acyclic Orientations of Graphs: Algebra, Geometry and Probability by Benjamin Iriarte Giraldo Submitted to the Department of Mathematics on February 25, 2015, in partial fulfillment of the requirements for the degree of Doctor of Philosophy in Mathematics

Abstract This thesis studies aspects of the set of acyclic orientations of a simple undirected graph. Acyclic orientations of a graph may be readily obtained from bijective labellings of its vertex-set with a totally ordered set, and they can be regarded as partially ordered sets. We will study this connection between acyclic orientations of a graph and the theory of linear extensions or topological sortings of a poset, from both the points of view of poset theory and enumerative combinatorics, and of the geometry of hyperplane arrangements and zonotopes. What can be said about the distribution of acyclic orientations obtained from a uniformly random selection of bijective labelling? What orientations are thence more probable? What can be said about the case of random graphs? These questions will begin to be answered during the first part of the thesis. Other types of labellings of the vertex-set, e.g. proper colorings, may be used to obtain acyclic orientations of a graph, as well. Motivated by our first results on bijective labellings, in the second part of the thesis, we will use eigenvectors of the Laplacian matrix of a graph, in particular, those corresponding to the largest eigenvalue, to label its vertex-set and to induce partial orientations of its edge-set. What information about the graph can be gathered from these partial orientations? Lastly, in the third part of the thesis, we will delve further into the structure of acyclic orientations of a graph by enhancing our understanding of the duality between the graphical zonotope and the graphical arrangement with the lens of Alexander duality. This will take us to non-crossing trees, which arguably vastly subsume the combinatorics of this geometric and algebraic duality. We will then combine all of these tools to obtain probabilistic results about the number of acyclic orientations of a random graph, and about the uniformly random choice of an acyclic orientation of a graph, among others. Thesis Supervisor: Richard P. Stanley Title: Professor of Applied Mathematics

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Acknowledgments During my years at MIT, I have had the fortune to be supervised by the same person whose mathematics once inspired me to become a mathematician in the first place. I am thankful to Richard P. Stanley for the obvious reasons why a student should be thankful to his/her advisor, but far more importantly, for his own mathematical work, which not only has inspired me, but also a vast number of people in the world. No mathematician on earth, at this stage of his/her career, can have the luxury to not acknowledge the contributions to their own scientific development of a large number of people. To get to this point, people need other people who inspire them. This is specially true in mathematics, where our job demands from us to regularly "steal" someone else's beautiful ideas and use them to generate other new ideas. Some of these sets of ideas that inspired me and that are present in this thesis, and their authors/divulgers, need to be thanked for. I'd like to thank the following mathematicians: Richard P. Stanley, Gian-Carlo Rota, Christos A. Athanasiadis, Bernd Sturmfels, Alexander Postnikov, Persi Diaconis, Ezra Miller, Tibor Gallai, Victor Guillemin, James Munkres, Arthur Engel, Roger Heath-Brown, Colin McDiarmid, Bela Bollobas, JeanPierre Serre, Matthias Beck, C.F. Gauss and the compass, and Oscar Bernal. People then need teachers who can help them bring their inspiration into action, and this transition is crucial. In my case, special thanks must go to Federico Ardila, Richard P. Stanley, Jacob Fox, Maricarmen Martinez, Jean Carlos Cortissoz, Serkan Hosten, and the members of my thesis comittee: Michelle Wachs and Henry Cohn. In order to make the best out of themselves, people also need colleagues who always give the best and who always ask for the best. I need to thank here my own share of excellent co-workers: Pablo Garcia, Pedro Rangel, Carlos Coelho, Brad Chase, Alejandro H. Morales, Michael Donovan, Joel B. Lewis, Luis G. Serrano, Haotian

Pang, and Alex Fink. Because there is always the risk of burnouts, people need to have good roots to keep the hard work at safe levels. My sincere thanks must go here to my family, Valeria Rueda, Stefan Bbssinger, Juan Camilo Velasquez, Andr6s D. Jaramillo, Janeth Velasco, Alejandro H. Morales, Pablo Garcia, Natalia Duque, Andr6s Cubillos, Alejandra Falla, and Pedro Rangel. And even with all these weapons at hand, things might just go wrong and life can get complicated. That's why people need to surround themselves with individuals of "magical power". These are the ones that, due to some uncanny wisdom and knowhow, can change the course of your professional life in a couple of conversations. Very special thanks to Federico Ardila, Andr6s Villaquiran, Sergey Fomin, Jacob Fox, Bernd Sturmfels, and Carly Klivans. Now, mathematics or any kind of work is a human effort and therefore, is bound to our human condition; we humans need support, hope, and light. Special thanks go to my mother, Maria Amparo Giraldo, whom I greatly admire. Lastly, people need to have a reason why. My greatest thanks then go to my wife, to my son, and to God.

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Contents Preface

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1

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Linear Extensions. 1.1 Introduction. . . . . . . . . . . . . . . . . . . . 1.2 Introductory results. . . . . . . . . . . . . . . . 1.2.1 The case of bipartite graphs. . . . . . . . 1.2.2 O dd cycles. . . . . . . . . . . . . . . . . 1.3 Comparability graphs. . . . . . . . . . . . . . . 1.3.1 G eom etry. . . . . . . . . . . . . . . . . . 1.3.2 Poset theory. . . . . . . . . . . . . . . . 1.4 Beyond comparability and enumerative results . 1.4.1 A useful technique. . . . . . . . . . . . . 1.4.2 General bounds for the main statistic. . 1.4.3 Random graphs . . . . . . . . . . . . . . 1.5 Further techniques. . . . . . . . . . . . . . . . .

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Largest Eigenvalue of the Laplacian Matrix. 2.1 Introduction. . . . . . . . . . . . . . . . . . . 2.2 Background and definitions. . . . . . . . . . . 2.2.1 The graphical arrangement. . . . . . . 2.2.2 Modular decomposition. . . . . . . . . 2.2.3 Comparability graphs. . . . . . . . . . 2.2.4 Linear algebra. . . . . . . . . . . . . . 2.2.5 Spectral theory of the Laplacian. . . . 2.3 Largest Eigenvalue of a Comparability Graph. 2.4 A characterization of comparability graphs . .

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Spanning Trees. 3.1 Introduction. . . . . . . . . . . . . . . . . . 3.2 Polytopal complexes for acyclic orientations. 3.2.1 A Classical Polytope. . . . . . . . . . 3.2.2 One More Degree of Freedom. . . . . 3.3 Two ideals for acyclic orientations. . . . . . 3.4 Non-crossing trees. . . . . . . . . . . . . . . 3.4.1 Standard monomials of TG . . . . . . . 6

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3.4.2 Non-crossing partitions. . . . . . . . . . . . . . A pplications. . . . . . . . . . . . . . . . . . . . . . . . 3.5.1 Random Acyclic Orientations of a Simple Graph: 3.5.2 Acyclic Orientations of a Random Graph. . . . 3.5.3 k-Neighbor Bootstrap Percolation. . . . . . . .

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. . . . . . . . . . . . . . . . Markov Chains. . . . . . . . . . . . . . . . .

82 85 85 92 94

List of Figures 1-1

1-2

2-1

3-1 3-2 3-3 3-4 3-5

3-6 3-7

An example of the function E for the case of bipartite graphs. Squares show the numbers that will be flipped at each step, and dashed arrows indicate arrows whose orientation still needs to be reversed. . . . . . . An example of the function 8' for the case of odd cycles. Squares show the numbers that will be flipped at each step. Dashed arrows indicate arrows whose orientation still needs to be reversed, while dashed-dotted arrows indicate those whose orientation will never be reversed. In particular, 4 will remain labeling the same vertex during all steps. . . (2-1a) Hasse diagram of a poset P on ground set [8]. (2-1b) Comparability graph G = G([8], E) of the poset P, where the closed regions depict the maximal proper modules of G. (2-1c) Unit eigenvector x E EAmax of G fully calculated, where dim (Ea) = 1. Arrows represent the induced orientation Ox of G. Notice the relation between . . . . . . . . . . . . . . . . OX, the modules of G, and the poset P. Examples of p.a.o.'s and the order relation i of Definition 3.2.11. . . Visual aids/guides to the proofs of (3-2a) Proposition 3.2.23 and (3-2b) Proposition 3.2.20.1. (3-2a) also offers an example for Definition 3.2.21. An example to the proof of Claim i.2 in Theorem 3.2.25. . . . . . . Example of a planar depiction, according to Definition 3.4.2. . . . . . Fully worked out example illustrating the central dogma of Section 3.4. Theorems 3.4.13 and 3.4.22 are dwelled on in tables 3-5e.i and 3-5e.ii, respectively, and in particular, fa = P. . . . . . . . . . . .. . Example of the bijection of Proposition 3.4.15. The selected spanning tree of Gr (in red) corresponds to the acyclic orientation 0 of G presented. Examples of Definitions 3.5.9 and 3.5.12 for the 4-cycle C4 . In 3-7a, we present the 1-skeleton of the graphical zonotope of C4, a rhombic dodecahedron, where the Cover-Reversal random walk runs; notably, it is not a regular graph. If four diagonals are added to the graph as shown in 3-7b, we obtain a 4-regular graph, AO"er in Proposition 3.5.14, where the Interval-Reversal random walk runs. . . . . . . . . . . . . .

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38 56 62 65 75

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Preface Two algorithmic problems have been in our mind since, approximately, the summer of 2013. A simplified and more restrictive version of the first problem is the following: 1 Problem 1. Let n c P. Call a sequence d = (di, d 2 ,..., d,) E NM" a graphical sequence if there exists a simple undirected graph G = G([n], E) such that, for all i E [n], the degree of vertex i in G is equal to di. In other words, d is a graphical sequence if it is the degree sequence of a simple undirected graph on vertex-set [n]. Suppose now that d E N" is a graphical sequence. Then, can we find an "efficient" and "transparent" algorithm to select, uniformly at random, a simple undirected graph on vertex-set [n] with degree sequence d?

The second problem is the following: Problem 2. Let n E P and let G = G([n], E) be a simple undirected graph. Call a map 0 : E -+ [n] x [n] such that 0(e) E {(i, j), (j,i)} for all e := {i, j} E E, an acyclic orientation of E or G, if the directed-graph on vertex-set [n] and directed-edge set O[E] has no directed-cycles. Generally, G possesses an exponential (on n) number of different acyclic orientations. But then, can we find an "efficient" and "transparent" algorithm to select, uniformly at random, an acyclic orientation of G? Perhaps, the reader might argue, these problems may be solvable in practice today by masterfully combining difficult tools from modern mathematical technology. We, reluctantly, would tend to be dissatisfied with these modern solutions and argue that instead, they still remain short in discovering and exploiting the fundamental properties of graphical sequences and/or of acyclic orientations and are, therefore, prone to be suboptimal. By not requiring our solutions to these problems to be "transparent", not only are we missing the opportunity to discover new mathematical theories, but also the opportunity to make these discoveries useful to outsiders. An inspiring example here to illustrate our point (to name at least one) are the surprising results of Broder {1989}. In that paper, the author introduced yet another method to select, uniformly at random, a spanning tree of a connected simple undirected graph. This thesis is a by-product of our efforts to solve Problem 2, and it presents three of the projects that we pursued in that vein. These three building blocks of the manuscript are, however, of independent mathematical interest and as such, they will be presented as self-contained units that are not only independent from Problem 2, but also from each other. During the first part, in Chapter 1, we will study the following problem:

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Given an underlying undirected simple graph, consider the set of all acyclic orientations of its edges. Each of these orientations induces a partial order on the vertices of our graph, and we can count the number of linear extensions of these posets. Then, what choice of acyclic orientation maximizes the number of linear extensions of the corresponding poset? This problem will be solved essentially for comparability graphs and odd cycles, presenting several proofs. The corresponding enumeration problem for arbitrary simple graphs will be studied, including the case of random graphs; this will culminate in 1) new bounds for the volume of the stable polytope and 2) strong concentration results for our main statistic and for the graph entropy, which hold true a.s. for random graphs. We will then argue that this problem springs up naturally in the theory of graphical arrangements and graphical zonotopes. In Chapter 2, we will study the eigenspace of the Laplacian matrix of a simple graph corresponding to the largest eigenvalue, subsequently arriving at the theory of modular decomposition of T. Gallai. Lastly, in Chapter 3, we will introduce three polytopal cell complexes associated with partial acyclic orientations of a simple graph, which generalize acyclic orientations. Using the theory of cellular resolutions, we will observe that two of these polytopal cell complexes minimally resolve certain special combinatorial polynomial ideals related to acyclic orientations. These ideals will explicitly be found to be Alexander dual, which relative to comparable results in the literature, generalizes in a cleaner and more illuminating way the well-known duality between permutohedron and tree ideals. The combinatorics underlying these results will then naturally lead us to a canonical way to represent rooted spanning forests of a labelled simple graph as non-crossing trees, and these representations will be observed to carry a plethora of information about generalized tree ideals and acyclic orientations of a graph, and about non-crossing partitions of a totally ordered set. We will then study in detail a small sample of the enumerative and structural consequences of collecting and organizing this information. Applications of this combinatorial miscellany will then be introduced and explored, namely: Stochastic processes on state space equal to the set of all acyclic orientations of a simple graph, including irreducible Markov chains, which exhibit stationary distributions ranging from linear extensions-based to uniform; a surprising formula for the expected number of acyclic orientations of an Erdbs-Renyi random graph; and a purely algebraic presentation of the main problem in bootstrap percolation, likely making it tractable to explore the set of all percolating sets of a graph with a computer.

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Chapter 1 Linear Extensions. 1.1

Introduction.

Linear extensions of partially ordered sets have been the object of much attention and their uses and applications remain increasing. Their number is a fundamental statistic of posets, and they relate to ever-recurring problems in computer science due to their role in sorting problems. Still, many fundamental questions about linear extensions are unsolved, including the well-known 1/3-2/3 Conjecture. Efficiently enumerating linear extensions of certain posets is difficult, and the general problem has been found to be OP-complete in Brightwell and Winkler {1991}. Acyclic directed-graphs, and similarly, acyclic orientations of simple undirected graphs, are closely related to posets, and their problem-modeling values in several disciplines, including the biological sciences, needs no introduction. On this chapter, we study the following problem: Problem 1.1.1. Suppose that there are n individuals with a known contagious disease, and suppose that we know which pairs of these individuals were in the same location at the same time. Assume that at some initial points, some of the individuals fell ill, and then they started infecting other people and so forth, spreading the disease until all n of them were infected. Then, assuming no other knowledge of the situation, what is the most likely way in which the disease spread out? Suppose that we have an underlying connected undirected simple graph G = G(V, E) with n vertices. If we first pick uniformly at random a bijection f : V -+ [n], and then orient the edges of E so that for every {u, v} E E we select (u, v) (read u directed to v) whenever f(u) < f(v), we obtain an acyclic orientation of E. In turn, each acyclic orientation induces a partial order on V in which u < v if and only if there is a directed-path (u, u1 ), (ui, u2 ), .. . , (uk, v) in the orientation. In general, several choices of f above will result in the same acyclic orientation. However, the most likely acyclic orientations so obtained will be the ones whose induced posets have the maximal number of linear extensions, among all posets arising from acyclic orientations of E. Our problem then becomes that of deciding which acyclic orientations of E attain this optimality property of maximizing the number of

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linear extensions of induced posets. This problem, referred to throughout this chapter as the main problem for G, was raised by Saito {2007} for the case of trees, yet, a solution for the case of bipartite graphs had been obtained already by Stachowiak {1988}. The main problem brings up the natural associated enumerative question: For a graph G, what is the maximal number of linear extensions of a poset induced by an acyclic orientation of G? This statistic for simple graphs will be herein referred to as the main statistic (Definition 1.3.12). The central goal of this initial chapter on the subject will be to begin a rigorous study of the main problem from the points of view of structural and enumerative combinatorics. We will introduce 1) techniques to find optimal orientations of graphs that are provably correct for certain families of graphs, and 2) techniques to estimate the main statistic for more general classes of graphs and to further understand aspects of its distribution across all graphs. In Section 1.2, we will present an elementary approach to the main problem for both bipartite graphs and odd cycles. This will serve as motivation and preamble for the remaining sections. In particular, in Section 1.2.1, a new solution to the main problem for bipartite graphs will be obtained, different to that of Stachowiak {1988} in that we explicitly construct a function that maps injectively linear extensions of non-optimal acyclic orientations to linear extensions of an optimal orientation. As we will observe, optimal orientations of bipartite graphs are precisely the bipartite orientations (Definition 1.2.1). Then, in Section 1.2.2, we will extend our solution for bipartite graphs to odd cycles, proving that optimal orientations of odd cycles are precisely the almost bipartite orientations (Definition 1.2.7). In Section 1.3, we will introduce two new techniques, one geometrical and the other poset-theoretical, that lead to different solutions for the case of comparability graphs. Optimal orientations of comparability graphs are precisely the transitive orientations (Definition 1.3.2), a result that generalizes the solution for bipartite graphs. The techniques developed on Section 1.3 will allow us to re-discover the solution for odd cycles and to state inequalities for the general enumeration problem in Section 1.4. The recurrences for the number of linear extensions of posets presented in Corollary 1.3.11 had been previously established in Edelman et al. {1989} using promotion and evacuation theory, but we will obtain them independently as by-products of certain network flows in Hasse diagrams. Notably, Stachowiak {1988} had used some instances of these recurrences to solve the main problem for bipartite graphs. Further on, in Section 1.4, we will also consider the enumeration problem for the case of random graphs with distribution G , 0 < p < 1, and obtain tight concentration results for our main statistic, across all graphs. Incidentally, this will lead to new inequalities for the volume of the stable polytope and to a very strong concentration result for the graph entropy (as defined in Csiszir et al. {1990}), which hold a.s. for random graphs. Lastly, in Section 1.5, we will show that the main problem for a graph arises naturally from the corresponding graphical arrangement by asking for the regions with maximal fractional volume (Proposition 1.5.2). More surprisingly, we will also observe that the solutions to the main problem for comparability graphs and odd cycles correspond to certain vertices of the corresponding graphical zonotopes (The-

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orem 1.5.3). Convention 1.1.2 (For Chapter 1). Let G = G(V, E) be a simple undirected graph. Formally, an (complete) orientation 0 of E or of G is a map 0 : E -+ V 2 such that for all e := {u, v} E E, we have 0(e) E {(u, v), (v, u)}. Furthermore, 0 is said to be acyclic if the directed-graph on vertex-set V and directed-edge-set O[E] is acyclic. On numerous occasions, we will somewhat abusively also identify an acyclic orientation 0 of E with the set O[E], or with the poset that it induces on V, doing this with the aim to reduce extensive wording. When defining posets herein, we will also try to make clear the distinction between the ground set of the poset and its order relations.

1.2 1.2.1

Introductory results. The case of bipartite graphs.

The goal of this section is to present a combinatorial proof that the number of linear extensions of a bipartite graph G is maximized when we choose a bipartite orientation for G. Our method is to find an injective function from the set of linear extensions of any fixed acyclic orientation to the set of linear extensions of a bipartite orientation, and then to show that this function is not surjective whenever the initial orientation is not bipartite. Throughout the section, let G be bipartite with n > 1 vertices. Definition 1.2.1. Suppose that G = G(V, E) has a bipartition V = V1 Li V2 . Then, the orientations that either choose (v 1 , v 2 ) for all {v 1 , v 2 } E E with v 1 E V1 and v 2 G V2 , or (v 2 7 v 1 ) for all {v 1 ,v 2 } E E with v 1 E V and v 2 E V2 , are called bipartite orientations of G. Definition 1.2.2. For a graph G on vertex-set V with |VI Bij(V, [n]) the set of bijections from V to [n].

=

n, we will denote by

As a training example, we consider the case when we transform linear extensions of one of the bipartite orientations into linear extensions of the other bipartite orientation. We expect to obtain a bijection for this case. Proposition 1.2.3. Let G = G(V, E) be a simple connected undirected bipartite graph, with n = |VI. Let Odown and 0,, be the two bipartite orientationsof G. Then, there exists a bijection between the set of linear extensions of Odon and the set of linear extensions of OU. Proof. Consider the automorphism rev of the set Bij(V, [n]) given by rev(f)(v) = n+ 1- f(v) for all v E V and f E Bij(V, [n]). It is clear that (rev o rev)(f) = f. However, since f(u) > f(v) implies rev(f)(u) < rev(f)(v), then rev reverses all directed-paths in any f-induced acyclic orientation of G, and in particular the restriction of rev to the set of linear extensions of Odown has image O,,P, and viceversa.

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We now proceed to study the case of general acyclic orientations of the edges of G. Even though similar in flavour to Proposition 1.2.3, our new function will not in general correspond to the function presented in the proposition when restricted to the case of bipartite orientations. To begin, we define the main automorphisms of Bij(V, [n]) that will serve as building blocks for constructing the new function. Definition 1.2.4. Consider a simple graph G = G(V, E) with |VI = n. For different vertices u, v E V, let revu, be the automorphism of Bij(V, [n]) given by the following rule: For all f G BIj(V, [n]), let

reVUV (f)(u) :reVU"(f)(V) revUV(f)(w)

= f (v), = f (U), = f(w) if w

c

V\{u,v}.

It is clear that (revuv o revuv)(f) = f for all f E Bij(V, [n]). Moreover, we will need the following technical observation about revuv. Observation 1.2.5. Let G = G(V, E) be a simple graph with |VI = n and consider a bijection f G Bij(V, [n]). Then, if for some u,v,x, y E V with f(u) < f(v) we have

that revu,(f)(x) > revec(f)(y) but f(x) < f(y), then f(u)

1 we know Ok, Bk and fk, and we want to compute Ok+1, Bk+1 and fk+1. If Bk = 0, then Ok = Oup and fk is a linear extension of Oup, so we stop our recursive process. If not, then Bk contains elements uk and vk such that fk(uk) and

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fk(vk) are respectively minimum and maximum elements of fk(Bk) 9 [n]. Moreover, 0 k+1 be the acyclic orientation of G Uk # Vk. We will then let fk+1 := revukVk(fk),

induced by fk+1, and calculate Bk+1 from Ok+1If we let m be the minimal positive integer for which Bm+ = 0, then 8(f) = (revumvm o ... o revU 2 V 2 o revulv1 ) (f). The existence of m follows from observing that Bk+1 C Bk whenever Bk 74 0. In particular, if Bk # 0, then Uk, Vk E Bk\Bk+1 and

so 1 < m < [IB1IJ. It follows that the pairs {{'uk, Vk}ke[m

are pairwise disjoint,

for all k c [m], and f(ui) < f(u 2 ) < -

< f(Um) < f(Vm) < ... < f(v 2 ) < f(vi). As a consequence, the automorphisms in the composition description of 9 commute. Lastly, fm+1 will be a linear extension of O" and we stop the inductive process by defining 8(f) = fm+i. To prove that 9 is injective, note that given 0 and fm+i as above, we can recover uniquely f by imitating our procedure to find 9(f). Firstly, set g, := fm+1 and Q, := OUP, and compute C1 C V as the set of vertices incident to an edge whose orientation differs in Q, and 0. Assuming prior knowledge of Qk, Ck and gA, and whenever Ck 74 0 for some positive integer k, find the elements of Ck whose images under A are maximal and minimal in gk(Ck). By the discussion above and Observation 1.2.5, we check that these are respectively and precisely Uk and Vk. Resembling the previous case, we will then let gk+1 := revukVk(gk), Qk+1 be the acyclic orienation of G induced by 9A, and compute Ck+1 accordingly as the set of vertices incident to an edge with different orientation in Qk+1 and 0. Clearly gm+1 = f, and the procedure shows that 9 is invertible in its image. To establish that 9 is not surjective whenever 0 74 Odown, note that then 0 contains a directed-2-path (w, u) and (u, v). Without loss of generality, we may assume that the orientation of these edges in 0, is given by (w, u) and (v, u). But then, a linear extension g of O,, in which g(u) = n and g(v) = 1 is not in Im(8) since otherwise, using the notation and framework discussed above, there would exist different i, j E [m] such that ui = u and vj = v, which then contradicts the choice of u1 and v 1. This completes the proof. f(uk) = fk(uk) and f(vk)

1.2.2

=

fk(vk)

Odd cycles.

In this section G = G(V, E) will be a cycle on 2n + 1 vertices with n > 1. The case of odd cycles follows as an immediate extension of the case of bipartite graphs, but it will also be covered under a different guise in Section 1.4. As expected, the acyclic orientations of the edges of odd cycles that maximize the number of linear extensions resemble as much as possible bipartite orientations. This is now made precise. Definition 1.2.7. For an odd cycle G = G(V, E), we say that an ayclic orientationof its edges is almost bipartite if under the orientation there exists exactly one directed2-path, i.e. only one instance of (u,v) and (v,w) in the orientation with u,v,w E V.

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4

2

Figure 1-1: An example of the function 8 for the case of bipartite graphs. Squares show the numbers that will be flipped at each step, and dashed arrows indicate arrows whose orientation still needs to be reversed. Theorem 1.2.8. Let G = G(V, E) be an odd cycle on 2n + 1 vertices with n > 1. Then, the acyclic orientations of E that maximize the number of linear extensions are the almost bipartite orientations. First proof. Since the case when n = 1 is straightforward let us assume that n> 2 and consider an arbitrary acyclic orientation 0 of G. Again, our method will be to construct an injective function e' that transforms every linear extension of 0 into a linear extension of some fixed almost bipartite orientation of G, where the specific choice of almost bipartite orientation will not matter by the symmetry of G. To begin, note that there must exist a directed-2-path in 0, say (u, v) and (v, w) for some u, v, w E V. Our goal will be to construct e' so that it maps into the set of linear extensions of the almost bipartite orientation Ov,, in which our directed-path (u, v), (v, w) is the unique directed-2-path. To find 8', first consider the bipartite graph G' with vertex-set V\{v} and edge-set E\ ({u, v} U {v, w}) U {u, w}, along with the orientation 0' of its edges that agrees on common edges with 0 and contains (u, w). Clearly 0' is acyclic. If f is a linear extension of 0, we regard the restriction f' of f to V\{v} as a strict order-preserving map on 0', and analogously to the proof of Theorem 1.2.6, we can transform injectively f' into a strict order-preseving map

g' with Im(g') = Im(f') = Im(f)\{f(v)} of the bipartite orientation of G' that contains (u, w). Now, if we define g E Bij(V, [n]) via g(x) = g'(x) for all x E V\{v} and g(v) = f(v), we see that g is a linear extension of Ouv.. We let E'(f) = g. The technical work for proving the general injectiveness of E', and its non-surjectiveness when 0 is not almost bipartite, has already been presented in the proof of Theorem 1.2.6: That 0' is injective follows from the injectiveness of the map transforming f' into g', and then by noticing that f(v) = g(v). Non-surjectiveness follows from noting that if 0 is not almost bipartite, then 0 contains a directed-2-path (a, b), (b, c)

18

5.

f 3

2

2

5 g

Figure 1-2: An example of the function 0' for the case of odd cycles. Squares show the numbers that will be flipped at each step. Dashed arrows indicate arrows whose orientation still needs to be reversed, while dashed-dotted arrows indicate those whose orientation will never be reversed. In particular, 4 will remain labeling the same vertex during all steps. with a, b, c E V and b = v, so we cannot have simultaneously g'(a) = min Im(f') and g'(c) = max Im (f').

1.3

Comparability graphs.

In this section, we will study our main problem of the chapter using more general techniques. As a consequence, we will be able to understand the case of comparability graphs, which includes bipartite graphs as a special case. Let us first recall the main object of this section: Definition 1.3.1. A comparability graph is a simple undirected graph G = G(V, E) for which there exists a partialorder on V under which two different vertices u, v E V are comparable if and only if {u, v} E E. The acyclic orientations of the edges of a comparability graph G that maximize the number of linear extensions are precisely the orientations that induce posets whose comparability graph agrees with G. Comparability graphs have been largely discussed in the literature, mainly due to their connection with partial orders and because they are perfectly orderable graphs and more generally, perfect graphs. Comparability graphs, perfectly orderable graphs and perfect graphs are all large hereditary classes of graphs. In Gallai's fundamental work {1967}, a characterization of comparability graphs in terms of forbidden subgraphs was given and the concept of modular decomposition of a graph was introduced.

19

Note that, given a comparability graph G = G(V, E), we can find at least two partial orders on V induced by acyclic orientations of E whose comparability graphs (obtained as discussed above) agree precisely with G, and the number of such posets depends on the modular structure of G. Let us record this idea in a definition. Definition 1.3.2. Let G = G(V, E) be a comparabilitygraph, and let 0 be an acyclic orientation of E such that the comparability graph of the partial order of V induced by 0 agrees precisely with G. Then, we will say that 0 is a transitive orientation of

G. We will present two methods for proving our main result. The first one (Subsection 1.3.1) relies on Stanley's transfer map between the order polytope and the chain polytope of a poset, and the second one (Subsection 1.3.2) is made possible by relating our problem to network flows.

1.3.1

Geometry.

To begin, let us recall the main definitions and notation related to the first method. Definition 1.3.3. We will consider R 1"] with euclidean topology, and let {ej}E[,l] be the standard basis of RH"]. For J C [n], we will define ej := E ,ej and eo := 0;

furthermore, for x c R nI we will let xj := ZjEJ x3 and xO := 0. Definition 1.3.4. Given a partial order P on [n), the order polytope of P is defined as: 0 (P) := {x E RI'n : 0 < xi

1 and xj 5

xk

whenever j

p k, V i, j, k E [n]}.

The chain polytope of P is defined as:

C (P) := {x E RI'I : xi > 0, V i E [n] and XC

K

1 whenever C is a chain in P}.

Stanley's transfer map cJ : 0 (P) -+ C (P) is the function given by: Cfx)

if i is not minimal in P,

xi - maxjoi x3

if i is minimal in P.

xi

Let P be a partial order on [n]. It is relatively simple to see from the definitions that the vertices of 0 (P) are given by all the ej with I an order filter of P, and those of C (P) are given by all the eA with A an antichain of P. Now, a well-known result of Stanley {1986} states that Vol (0 (P)) = Le(P) where e(P) is the number of linear extensions of P. This result can be proved by considering the unimodular triangulation of 0 (P) whose maximal (closed) simplices x 1(n) _< 1} with X,-1( 2 ) < ... have the form A, := {x E RIn] : 0 < X,-1(i) u : P -+ n a linear extension of P. However, the volume of C (P) is not so direct to compute. To find Vol (C (P)), Stanley made use of the transfer map 4, a pivotal idea that we now wish to describe in detail since it will provide a geometrical point of view on our main problem.

20

It is easy to see that 4) is invertible and its inverse can be described by:

4r'(x)i =

max

C chain in P: i is maximal in C

xC, for all i e [n] and x E C (P).

As a consequence, we see that dr 1 (eA) = eAv for all antichains A of P, where AV is the order filter of P induced by A. It is also straightforward to notice that 4) is linear on each of the A, with u a linear extension of P, by staring at the definition of A,. Hence, for fixed u and for each i E [n], we can consider the order filters AY := ou-([i, n]) along with their respective minimal elements Ai in P, and notice that 4)(eAY) = eA, and also that ct(0) = 0. From there, 4) is now easily seen to be a unimodular linear map on A., and so Vol (4) (A,)) = Vol (A,) = 3. Since 4) is invertible, without unreasonable effort we have obtained the following central result: Theorem 1.3.5 (Stanley {1986}). Let P be a partialorder on [n]. Then, Vol(0 (P)) = Vol(C (P)) = le(P), where e(P) is the number of linear extensions of P. Definition 1.3.6. Given a simple undirected graph G

=

G([n], E), the stable poly-

tope STAB(G) of G is the full dimensional polytope in R [ obtained as the convex hull of all the vectors e 1 , where I is a stable (a.k.a. independent) set of G. Now, the chain polytope of a partial order P on [n] is clearly the same as the stable polytope STAB (G) of its comparability graph G = G([n], E) since antichains of P correspond to stable sets of G. In combination with Theorem 1.3.5, this shows that the number of linear extensions is a comparability invariant, i.e. two posets with isomorphic comparability graphs have the same number of linear extensions. We are now ready to present the first proof of the main result for comparability graphs. We will assume connectedness of G for convenience in the presentation of the second proof. Theorem 1.3.7. Let G = G(V, E) be a connected comparability graph. Then, the acyclic orientations of E that maximize the number of linear extensions are exactly the transitive orientations of G. First proof. Without loss of generality, assume that V = [n]. Let 0 be an acyclic orientation of G inducing a partial order P on [n]. If two vertices i, j E [n] are incomparable in P, then {i, j} V E. This implies that all antichains of P are stable sets of G, and so C (P) C STAB (G). On the other hand, if 0 is not transitive, then there exists two vertices k, f E [n] such that {k, f} V E, but such that k and t are comparable in P, i.e. the transitive closure of 0 induces comparability of k and f. Then, ek + et is a vertex of the stable polytope STAB (G) of G, but since C (P) is a subpolytope of the n-dimensional cube, ek + ee V C (P). We obtain that C (P) \$ STAB (G) if 0 is not transitive, and so

C (P) ; STAB (G). If 0 is transitive, then C (P) = STAB (G). This completes the proof.

21

1.3.2

Poset theory.

Let us now introduce the background necessary to present our second method. This will eventually lead to a different proof of Theorem 1.3.7. Definition 1.3.8. If we consider a simple connected undirected graph G = G(V, E) and endow it with an acyclic orientation of its edges, we will say that our graph is an oriented graph and consider it a directed-graph, so that every member of E is regarded as an ordered pair. We will use the notation Go = Go(V, E) to denote an oriented graph defined in such a way, coming from a simple graph G. Definition 1.3.9. Let Go = GO(V, E) be an oriented graph. We will denote by Go the oriented graph with vertex-set V := V U {, } and set of directed-edges E equal to the union of E and all edges of the form: (v,1) with v G V and outdeg (v) = 0 in Go, and (0, v) with v G V and indeg (v) = 0 in Go. A natural flow on Go will be a function f : E -+ N such that for all v E V, we have:

Z (xv)

f(x, v)=

E

f(v, y). (vy)

E

In other words, a natural flow on Go is a nonnegative integer network flow on Go with unique source 0, unique sink 1, and infinite edge capacities. First, let us relate natural flows on oriented graphs with linear extensions of induced posets. Lemma 1.3.10. Let Go = GO(V, E) be an oriented graph with induced partial order P on V, and with |VI = n. Then, the function g : E -+ N defined by

= I c: o- is a linear extension of P and a(u) = u(v) if (u, v) E E,

g(v,1)

=

-

g(u, v)

o is a linear extension of P and o(v)

if v g(0, v)

=

=

n}I

C V and outdeg (v) = 0 in Go, and

I{

: o- is a linear extension of P and u(v) = 1

if v E V and indeg (v) = 0 in Go, is a naturalflow on Go. Moreover, the net g-flow from 0 to 1 is equal to e(P). Proof. Assume without loss of generality that V = [n], and consider the directedgraph K on vertex-set V(K) = [n] U {0, i} whose set E(K) of directed-edges consists

of all: (i,j) for i

n (log2 logbn)" ] < logn

Combining these two results, we see that:

E[ log 2

(G)]

(nlog 2 s)(1 + o(1)), 28

H(G)

in (- log 2 a )

=

-

log2

n/ ai)

\

-

log 2 !oc(G) n

=

log 2

(

and moreover, that log 2 (G) ~ E[ log 2 e(G)] a.s. holds. The second necessary inequality comes, firstly, from using Observation 1.4.1, so that e(G) < n!Vol (STAB (G)), and then from a direct application of Theorem 1.4.10. log 2 e(G). Now, we further observe that for We obtain that n(log 2 n - H(G)) 1- y(G), and then: a E STAB (G), we have Ej -a.

s + c holds A classic result of Grimmett and McDiarmid {1975} states that X(G) a.s., where c =2 log b+1. Hence, a.s., H(G) (log 2 g)= log 2 n-log 2 (s+0(1)), and then nlog 2 (s + 0(1)) : log 2 E(G). From here, we directly obtain: log 2

F(G)

nlog2 s+O

(n)

(1.4.3)

a.s..

Therefore, from inequalities 1.4.2 and 1.4.3: log 2 F-(G) = nlog2 s+O(n) a.s..

Calculating inequality 1.4.3 more precisely by dropping the 0-notation and using Grimmett and McDiarmid's constant, we obtain:

n!

e

Sns"

2/(0og2 b)

n Vol(STAB(G)) < - . Cnj* a.s., where c = 2 n!2

Corollary 1.4.13. Let G ~ GnP with 0 < p < 1, b = n 2 logo logo n. Then, for large enough n: log 2

1.5

+ +

H(G) 5 log 2 (

+

and s = 2log n

-

(1n

logn

.

Sn

2

-

Corollary 1.4.12. Let G - G,, with 0 < p < 1, b = 11P and s = 2 logo logo n. Then, for large enough n:

a.s..

Further techniques.

In this section, we will see how the main problem of the chapter has two more presentations as selecting a region in the graphical arrangement with maximal fractional volume, or as selecting a vertex of the graphical zonotope that is farthest from the origin in Euclidean distance.

29

Definition 1.5.1. Consider a simple undirected graph G = G([n], E). The graphical arrangement of G is the central hyperplane arrangement in IRH1 given by: AG = Ix

C Rn] : xi - xj = 0 , for some {i, j}

E}.

The regions (see Definition 2.2.1 and the comments thereafter) of the graphical arrangement AG with G = G([n], E) are in one-to-one correspondence with the acyclic orientations of G (Proposition 2.2.2). Moreover, the complete fan in R[n] given by AG is combinatorially dual to the graphicalzonotope of G: Za"'''

=

[e

-

e3 , e

-

e1,

{i,j}EE

and there is a clear correspondence between the regions of AG and the vertices of Zntral.

Following Klivans and Swartz {2011}, we define the fractional volume of a region

R of AG to be: Vol' (R)

-

Vol (B n nR)

Vol (Bn)

where B" is the unit n-dimensional ball in R [n. With little work, using Proposition 2.2.2 and the symmetry of the permutohedron, it is possible to say the following about these volumes: Proposition 1.5.2. Let G = G([n], E) be an undirected simple graph, and let AG be its graphical arrangement. If R is a region of AG and P is its corresponding partial order on [n] under the map of Proposition 2.2.2, then:

e(P) n!

Volf (R)= n!

The problem of finding the regions of AG with maximal fractional volume is, intuitively, closely related to the problem of finding the vertices of Z* t ral that are farthest from the origin under some appropriate choice of metric. It turns out that, with Euclidean metric, a precise statement can be formulated when G is a comparability graph: Theorem 1.5.3. Let G = G(V, E) be a comparability graph. Then, the vertices of the graphical zonotope of Zcntral that have maximal Euclidean distance to the origin are precisely those that correspond to the transitive orientations of E, which in turn have maximal number (G) of linear extensions. To prove Theorem 1.5.3, we first note that for a simple (undirected) graph G = G(V, E), the vertex of Zn t a corresponding to a given acyclic orientation of E is precisely the point:

(outdeg (v) - indeg(v))vEV, where outdeg (.) and indeg (.) are calculated using the given orientation. We need to establish a preliminary lemma.

30

Lemma 1.5.4. Let Go = Go(V, E) be an oriented graph. Then, (indeg (v) - outdeg (v)) 2 = E + tri (Go) + incom (Go) - com (Go), vEV

where: 1. tri (Go) is the number of directed-triangles(u, v), (v, w), (u, w) E E. 2. incom(Go) is the number of triples u,v,w C V such that (v,w), (w,v) 0 E but

either (u, v), (u, w) E E or (v, u), (w, u) E E.

E E such that (u, w)

\$

3. com (Go) is the number of directed-2-paths (u, v), (v, w)

E. Proof. For v E V, outdeg (v) 2 is equal to outdeg (v) plus two times the number of pairs u / w such that (v, u), (v, w) E E, indeg (v) 2 is equal to indeg (v) plus two times the number of pairs u, 74 w such that (u, v), (w, v) E E, and outdeg (v) - indeg (v) is equal to the number of pairs u 74 w such that (u, v), (v, w) E E. If we add up these terms and cancel out terms in the case of directed-triangles, we obtain the desired equality.

An important consequence of Lemma 1.5.4 is the following: If G = G(V, E) is a simple graph, all the in their values of tri(-) and of |E|, which incom (.). Moreover, com (-) + incom (-) G of the form {u, v}, {v, w} E E with u choice of orientation for E.

acyclic orientations of E will not vary depend on G, but only in com (.) and is equal to the number of 2-paths in 74 w, so it is also independent of the

Proof of Theorem 1.5.3. We apply Lemma 1.5.4 directly. Since G is a comparability graph, from Theorem 1.3.7, we know that the value of incom (.) - com (-) will be maximized precisely on the transitive orientations of G, since all transitive orientations force com (-) = 0.

Remark 1.5.5 (To Theorem 1.5.3). In fact, per the Second proof of Theorem 1.2.8 in Subsection 1.4.1, the analogous result to Theorem 1.5.3 also holds true for odd cycles. In joint work with Kyle Gettig, we have again made use of Observation 1.4.2 to prove that odd anti-cycles (i.e. the complement graphs to odd cycles) also have this property.

31

32

Chapter 2 Largest Eigenvalue of the Laplacian Matrix. 2.1

Introduction.

Let G = G([n], E) be a simple (undirected) graph, where [n] = {1, 2,..., n}, n The adjacency matrix of G is the n x n matrix A = A(G) such that:

(

=

1

C P.

if {i,j} E E,

A The Laplacian matrix of G is the n x n matrix L = L(G) such that: (L)=

di

if i = j,

:

where di = dG(i)= (dG)i is the degree of vertex i in G. The spectral theory of these matrices, i.e. the theory about their eigenvalues and eigenspaces, has been an object of much study for the last 40 years. The roots of this beautiful theory, however, can arguably be traced back to Kirchhoff's matrixtree theorem, whose first proof is often attributed to Borchardt {1860} even though at least one proof was already known by Sylvester {1857}. A recollection of some of the interesting applications of the theory can be found in Spielman {2009}, and more complete accounts of the mathematical backbone are Brouwer and Haemers {2011} and Chung {1997}. Still, it would be largely inconvenient and prone to unfair omissions to attempt here a brief account of the many past and present contributors and contributing papers to the modern spectral theory of graphs, and we refer the reader to our references for further inquiries of the literature. This chapter aims to fill one (of the many) gap (s) in our present knowledge of the spectrum of the Laplacian matrix, namely, the lack of results about its eigenvectors with largest eigenvalue. We will answer the question: What information about the structure of a graph is carried in these eigenvectors? This question will be studied while remaining loyal to the central theme of this thesis: The mathematics of acyclic

33

axAmax(G),

be the (real) eigenvalues of L, and note that A 2 > 0 if G is a connected graph; we have effectively dropped G from the notation for convenience but remark that eigenvalues and eigenvectors depend on the particular graph in question, which will be clear from the context. We will also let EA, be the eigenspace corresponding to A. In its most primitive form, Fiedler's nodal domain's theorem {Fiedler, 1975} states that when G is connected and for all x e EA 2 , the induced subgraph G [{i E [n] : xi > 0}] is connected. Related ideas and results might be found in Merris {1998}. On this chapter, we will go even further in the way in which eigenvectors of the Laplacian may be used to learn properties of G. To explain this, let us firstly call a map,

0 :E -+ ([n] x [n]) U E

=

[n] 2 U E,

such that O(e) c {e, (i, j), (j, i)} for all e := {i, j} E E, an (partial) orientation of E (or G), and say that, furthermore, 0 is acyclic if O(e) \$ e for all e and the directed-graph on vertex-set [n] and edge-set O[E] has no directed-cycles. Along the exposition of the chapter, we will oftentimes also identify the object 0 with the set

0[E]. During this second part of the thesis, eigenvectors of the Laplacian and more precisely, elements of Ekmax, will be used to obtain orientations of certain (not necessarily induced) subgraphs of G. Henceforth, given G and for all x c R[n, the reader should always automatically consider the orientation (map) Ox = Ox(G) associated to x, Ox: E -+ [n] 2 U E, such that for e := {ij} c E: if Xi = X3

e

,

- -"n =

orientations of graphs. Our work follows the spirit of Fiedler {2011}, who pioneered the use of eigenvectors of the Laplacian matrix to learn about a graph's structure. One of the first observations that can be made about L is that it is positivesemidefinite, a consequence of it being a product of incidence matrices. We will thus let, 0 = A A2 < .

Ox(e) =

(i, j) if xi < xj, (j, i)

if Xi > Xj.

The orientation Ox will be said to be induced by x (e.g. Figure 2-1c). Implicit above is another subtle perspective that we will adopt, explicitly, that vectors x E RMn] are real functions from the vertex-set of the graph in question (all our graphs will be on vertex-set [n]). In our case, this graph is G, and even though accustomed to do so otherwise, entries of x should be really thought of as being indexed by vertices of G and not simply by positive integers. Later on in Section 2.3, for example, we will regularly state (combinatorial) results about the fibers of x when x belongs to a certain subset of RM"I (e.g. Emax), thereby regarding these fibers as vertex-subsets of the particular graph being discussed at that moment. Using this perspective, we will learn that the eigenspace ENx. is closely related to the theory of modular decomposition of Gallai {1967}; orientations induced by elements of EXax lead naturally to the discovery of modules. This connection will

34

most concretely be exemplified when G is a comparability graph, in which case these orientations iteratively correspond to and exhaust the transitive orientationsof G. It will be instructive to see Figure 2-1 at this point. In Section 2.2, we will introduce the background and definitions necessary to state the precise main contributions of this chapter. These punch line results will then be presented in Section 2.3. The central theme of Section 2.3 will be a stepwise proof of Theorem 2.3.1, our main result for comparability graphs, which summarily states that when G is a comparability graph, elements of Ex... induce transitive orientations of the copartition subgraph of G. It will be along the natural course of this proof that we present our three main results that apply to arbitrary simple

graphs: Propositions 2.3.10 and 2.3.11, and Corollary 2.3.12. Finally, in Section 2.4, we will present a curious novel characterization of comparability graphs that results from the theory of Section 2.3.

2.2

Background and definitions.

Let us first review the mathematical background relevant to this chapter. A few of the concepts that we will present here have already been introduced in Chapter 1, but we will restate them to free the reader from having to skip repeatedly over a large number of pages. In any case, we will also add small further clarifications to the original setting of Chapter 1, adapting the narrative of this chapter to fit our current needs.

2.2.1

The graphical arrangement.

Definition 2.2.1. Let G = G([n], E) be a simple (undirected) graph. The graphical arrangement of G is the union of hyperplanes in RlR: AG := {x G RI'l : xi - xj = 0 , for some {i,j} E E}.

Basic properties of graphical arrangements and, more generally, of hyperplane arrangements, are presented in Chapter 2 of Stanley {2004}. For G as in Definition 2.2.1, let R(AG) be the collection of all (open) connected components of the set RHl\Ac. An element of R(AG) is called a region of AG, and every region of AG is therefore an n-dimensional open convex cone in RI"1. Furthermore, the following is true about regions of the graphical arrangement: Proposition 2.2.2. Let G be as in Definition 2.2.1. Then, for all R x, y E R, we have that:

C R((AG)

and

OR:= OX - O. Moreover, the map R '-+ OR from the set of regions of AG to the set of orientations of E is a bijection between R(AG) and the set of acyclic orientations of G. Motivated by Proposition 2.2.2 and the comments before, we will introduce special notation for certain subsets of R "I obtained from AG-

35

Notation 2.2.3. Let G be as in Definition 2.2.1. For an acyclic orientation0 of E, we will let CO denote the n-dimensional closed convex cone in R 'n that is equal to the topological closure of the region of AG corresponding to 0 in Proposition2.2.2.

2.2.2

Modular decomposition.

We need to concur on some standard terminology and notation from graph theory, so let G = G([n], E) be a simple (undirected) graph and X a subset of [n]. As customary, C denotes the complement graph of G. The notation N(X) denotes the open neighborhood of X in G: N(X)

{j E [n]\X: there exists some i E X such that {i,j} E E}.

The induced subgraph of G on X is denoted by G[X], and the binary operation of graph disjoint union is represented by the plus sign +. Lastly, for Y C [n], X and Y are said to be completely adjacent in G if: X n Y = 0, and for all i E X and

j

E Y, we have that {i, j} E E.

The concepts of module and modular decomposition in graph theory were introduced by Gallai {1967} as a means to understand the structure of comparability graphs. The same work would eventually present a remarkable characterization of these graphs in terms of forbidden subgraphs. Section 2.3 of the present work will present an alternate and surprising route to modules. Definition 2.2.4. Let G = G([n], E) be a simple (undirected) graph. A module of G is a set A C [n] such that for all i, j G A: N(i)\A = N(j)\A

=

N(A).

Furthermore, A is said to be proper if A C [n], non-trivial if if G[A] is connected.

Al > 1, and connected

Corollary 2.2.5. In Definition 2.2.4, two disjoint modules of G are either completely adjacent or no edges exist between them. Let us now present some basic results about modules that we will need.

Lemma 2.2.6 (Gallai {1967}). Let G = ([n], E) be a connected graph such that Z is connected. If A and B are maximal (by inclusion) proper modules of G with A then A n B = 0.

\$

B,

Corollary 2.2.7 (Gallai {1967}). Let G = ([n], E) be a connected graph such that C is connected. Then, there exists a unique partitionof [n] into maximal propermodules of G, and this partition contains more than two blocks.

36

From Corollary 2.2.7, it is therefore natural to consider the partition of the vertexset of a graph into its maximal modules; the appropriate framework for doing this is presented in Definition 2.2.8. Hereafter, however, we will assume that our graphs are connected unless otherwise stated since (1) the results for disconnected graphs will follow immediately from the results for connected graphs, and (2) this will allow us to focus on the interesting parts of the theory.

Definition 2.2.8 (Ramirez-Alfonsin and Reed {2001}). Let G = G([n], E) be a connected graph. such that:

We will let the canonical partition of G be the set P

=

P(G)

a. If G is connected, P is the unique partition of [n] into the maximal proper modules of G. b. If C is disconnected, P is the partitionof [n] into the vertex-sets of the connected components of C. Hence, in Definition 2.2.8, every element of the canonical partition is a module of the graph. Elements of the canonical partition of a graph on vertex-set [8] are shown

in Figure 2-1b. Definition 2.2.9. In Definition 2.2.8, we will let the copartition subgraph of G be the graph GP on vertex-set [n] and edge-set equal to:

E\ {{i, j} E E: i, j E A for some A E P}.

2.2.3

Comparability graphs.

We had anticipated the importance of comparability graphs for the results of this chapter, so let us now recall what they are. Definition 2.2.10. A comparability graph is a simple (undirected) graph G = G(V, E) such that there exists a partial order on V under which two different vertices u, v E V are comparable if and only if {u, v} G E. A comparability graph on vertex-set [8] = {1, 2, . . , 8} is shown in Figure 2-1b. Comparability graphs are perfectly orderable graphs and more generally, perfect graphs. These three families of graphs are all large hereditary classes of graphs. Note that, given a comparability graph G = G(V, E), we can find at least two partial orders on V whose comparability graphs (obtained as discussed in Definition 2.2.10) agree precisely with G, and the number of such partial orders depends on the modular decomposition of G. Let us record this idea in a definition. Definition 2.2.11. Let G = G(V, E) be a comparabilitygraph, and let 0 be an acyclic orientation of E. Consider the partial order induced by 0 under which, for u, v E V, u is less than v iff there is a directed-path in 0 that begins in u and ends in v. If the comparability graph of this partial order on V (obtained as in Definition 2.2.10) agrees precisely with G, then we will say that 0 is a transitive orientation of G.

37

(c)

(b)

(a)

1 2 3 4 5 6

1

2

a

-0.1515... b = -0.2587... a c =-0.1021.. d = -0.1866... d

e0=.8855...

2

8

-a

Figure 2-1: (2-1a) Hasse diagram of a poset P on ground set [8]. (2-1b) Comparability graph G = G([8], E) of the poset P, where the closed regions depict the maximal proper modules of G. (2-1c) Unit eigenvector x E Exmax of G fully calculated, where dim (EAniax) = 1. Arrows represent the induced orientation Ox of G. Notice the relation between Ox, the modules of G, and the poset P.

2.2.4

Linear algebra.

Some standard terminology of linear algebra and other related conventions that we adopt are presented here. Firstly, we will always be working in Euclidean space RI'J, and all (Euclidean-normed real) vector spaces considered are assumed to live therein. Euclidean norm is denoted by 11 -I1. The standard basis of R[n] will be {ei}[], as customary. Generalizing this notation, for all I C [n], we will also let:

Zj:=Eei. iEI

The orthogonal complement in R["I to spanR (e[n]) will be of importance to us, so we will use special notation to denote it: R*[n] := (spanR

e[l]))-

For an arbitrary vector space V and a linear transformation T : V -+ V, we will say that a set U C V is invariant under T, or that T is U-invariant, if T(U) C U. Lastly, a key concept of this chapter: C[n],

For a vector x E Rc

and a set

C

we will say that E is a fiber of x if there exists oc E R such that xi = oc if and only if i E

4.

The notion of being a generic vector in a certain vector space, to be understood from the point of view of Lebesgue measure theory, is a central ingredient in many of our results. We now make this notion precise.

Definition 2.2.12. Let V be a linear subspace of R[n] with dim (V) > 0. We will say that a vector x E V is a uniformly chosen at random unit vector or u.c.u.v.

38

if x is

uniformly chosen at random from the set {y E V : IIyI = 1}. For x E V a u.c.u.v., a certain event or statement about x is said to occur or hold true almost surely if it is true with probability one.

2.2.5

Spectral theory of the Laplacian.

We will need only a few background results on the spectral theory of the Laplacian matrix of a graph. We present these below in a single statement, but refer the reader to Brouwer and Haemers {2011} for additional background and history. Lemma 2.2.13. Let G = G([n], E) be a simple (undirected) graph. Let L = L(G) - 0, z + e(x -p)

Q.

On the contrary, (x, z) 9 int ([P, and F > 0, such that z + (x - p) E

Q])

(

i y E int ([P, Q]) if and only if there exist p* E relint (P) and q* E Q such that y E (p*, q*), if and only if there exist p** C relint (P) and q** C relint (Q) such that y E (p**,q**).

if and only if there exists p E relint (P)

Q.

iii Let 7 arff(P) : R n) 4 R[n] be the projection operator onto the affine hyperplane containing P. If taff(P)[(x, z)]n relint (P)n relint (aff(P) [ 0, then (x, z) G

int ([P, QI). Proof. We will obtain these results in order. i (See also Figure 3-2b) We prove the "only if" direction for both statements. Suppose that y E int ([P, Q]) and let p C P and q c Q be such that y E (p, q). Let us assume that p E a (P). Take an open (n - 1)-dimensional ball By C

int ([P, Q]) centered at y such that aff(By) is parallel to aff(P) and aff(Q). Let 60

C be the positive open cone generated by By - q, and consider the affine open cone q + C. Then, B,, : (q + C) n aff(P) is an open (n - 1)-dimensional ball in aff(P) such that p E relint (B.). Hence, since P is also (n - 1)-dimensional, there exists some pi E relint (P) n B.. Now, let yi = (p1, q) n By E int ([P, Q]). Since Y2 := y + (y - yi) E B, 9 int ([P, Q]), there exist P2 E P and q2 E Q such that Y2 = (p2 , q2) n BY. But then, there exist p* E (P1, P2) C relint (P) and q* E (q, q2 ) g Q such that y = (*, q*) nBy, as we wanted. If q* E a (Q), we can now repeat an analogous construction starting from q* and p* to find p** E relint (P) and q** E relint (Q) such that y E (p**, q**). ii This is a consequence of i, and not easy to prove without it. We prove the second statement, which is equivalent to the first. For the "if" direction, suppose that for some p E relint (P) and e > 0, z + F(x - p) E Q. Take some y E (x, z) and consider the line containing both z + e(x - p) and y. For a sufficiently small e, this line intersects aff(P) in some pi E relint (P). But then, for a small open ball Bp, C relint (P) centered at p, and with aff(Bp 1 ) = aff(P), the open set (Bp 1 , z) contains y and lies completely inside int ([P, Q]), so y E int ([P, Q]). For the "only if" direction, suppose that (x, z) C int ([P, Q]) and take y E (x, z). If x E relint (P), then we are done since Q is also (n - 1)-dimensional. If x c a (P), from i, take p c relint (P), p # x, and q E Q with y E (p, q). But then, z + E(x - p) = q E Q for some e > 0. iii Take p E flafP) [(x, z)] n relint (P) n relint (7rtaffLp) [Q]) and let p = 0 be a normal to aff(P). Then, for some y E (x, z) and real number Oc : 0, y E (p, p + Ocp) and p + Ocp E relint (Q), so i shows that y E int ([P, Q]). Clearly then (x, z) C

int ([P, Q]).

,

Definition 3.2.21. Let G = G([n], E) be a simple graph, and let 0 be a p.a.o. of G with connected partition Y and acyclic orientation OF- of G'. Let us write Y0,in for the set of elements of E that are minimal in (E, o5 ), and for i E [n] with i E p E let:

IG(i, O) ={- E Z: a or p}. With this notation, we now define certain functions associated to 0 and G, called

height and depth: height , depthG : height(i)

[n]

-*

Q, 1

-

depthG(2) =

height(j).

jEIV(i,O) Example 3.2.22. Figure 3-2a exemplifies Definition 3.2.21 on a particular graph G on vertex-set [14] = {1, 2,... ,14}, with given p.a.o. 0. Since both heights

61

aff(P

0:

I

a 614414

'

5-"

b)aff

1 3

E ={0,U2,

3},/p=

heightg constant on each

{117, 1} ci. i E

depthG constant on each oa, i

E

161, e.g. height (6) [61, e.g. depthG(13)

=

1

=

(a) (b)

Figure 3-2: Visual aids/guides to the proofs of (3-2a) Proposition 3.2.23 and (3-2b) Proposition 3.2.20.i. (3-2a) also offers an example for Definition 3.2.21. are constant within each element/block of the connected partition E = {oa = {7}, -2 = {1, 2}, 0 3 = {6,10,14}, .., o-6 = {3, 4, 8}} associated to 0, we present only that common value for each block in the figure. and

depth'

1 >

depthG(i) >

.

Proposition 3.2.23. In Definition 3.2.21, let 9 E JG(O) and let p 9 [n] intersect every element of 5i~ni in exactly one point and contain only minimal elements of 0. Then: iEpn(T

if

and only if

= [n], and

.

Moreover, if G is connected, then iE np ePtho(i) > n depthGi) - E> whenever this holds,

Remark 3.2.24. Figure 3-2a shows one such choice of a set p in Proposition 3.2.23 that works for Example 3.2.22 (in red). Proof. The verification is actually a simple double-counting argument using the fact that o- is an order ideal, so we omit it. When G is connected, if o 5 [n], then there must exist i E [n]\ that is strictly greater in 0 than some element of u (and hence strictly greater than some element of p), again since u is an order ideal. Clearly, we must have heightg(i) >

-

Theorem 3.2.25. Let G = G([n], E) be a connected simple graph with abstract cell complex G as in Definition 3.2.18. For N > 0, N / n + E|, consider the (n 1)-dimensional simplex NA = cony (Nei, Ne 2 ,.... Nen) in R "n.If we let YG be

62

the polytopal complex obtained from the join [ZG, NA] after removing the (open) ndimensional cell and the (relatively open) (n - 1)-dimensional cell corresponding to NA, then YG is a polytopal complex realization of g(G. Proof. Let the faces of 3G obtained from 2H'1\{1 n1,o correspond to the faces of 0 (NA) in the natural way. Also, let the faces of XG of the form ([n], 0) correspond to the faces of ZG as in Theorem 3.2.15. The result is clearly true for the restriction to this two sub-complexes, so we will concentrate our efforts on the remaining cases. First, for the sake of having a lighter notation during the proof, we will let ' = [n]\p for any set p g [n]. A (relatively open) cell of YG\(ZG U a (NA)) can only be obtained as the strict join of a cell of 0 (ZG) and a cell of a (NA), so let us adopt some conventions to refer to this objects. Convention 3.2.26. During the course of the proof, we will let S (or So) denote a generic non-empty relatively open cell of NA obtained from p 9 [n] (resp. po), and F (or Fo) a generic relatively open cell of ZG with p. a. o. 0 of G, associated connected partition Y of G, and acyclic orientation 0 ' of G' yielding 0 (resp. 00, Eo, 0o1o). We argue that we will be done if we can prove the following claim: Claim i a) (F, S) is a cell of YG if and only if b) p = [n] and p is a non-empty union of elements from the set {u E E: u is maximal in (1, oz)}. When this equivalence is established, then we will let (F, S) correspond to the pair (', O3) E XG, where O1i denotes the restriction of 0 to E(G[]). Indeed, assume that Claim i holds. Then, under the stated correspondence of ground sets of cells, all elements of XG are uniquely accounted for as cells of YG. This is true for ZG clearly, and for the remaining cases since for any choice of a1 G [n], 5 0, and of p.a.o. 01 of G[u1], we can always extend uniquely 01 to a p.a.o. of G \$in which all the elements of 'ij are maximal. Secondly, we verify that -< corresponds to face containment in 9 G. Suppose that

(Fo, So) and (F, S) are cells of YG. Then, (Fo, So) 9 (F, S) if and only if Fo g F and So g 5, if and only if JG(O) C JG(OO) and po 9 p. Now, assuming Claim i, the last statement is true if and only if JG[p](OI) 9 JG[ -](0j): The difficult part here is the "if" direction. Clearly, po g p. Since p is a union of elements of Y that are maximal in (1, Icl + IE(G o])I= E= xI and x - p must have a negative entry in a. Therefore, z + (x - p) all E > 0 and Proposition 3.2.20.ii shows that (x, z) C 0 ([ZG, NA]).

'

NA for

Claim i.2 Let F0 , 00, 1o, 0", So, po be as in Convention 3.2.26. Then, there exist F, 0, T, O', S, p also as in Convention 3.2.26, such that p is a union of elements of the set {0 E Y : ux is maximal in (T, oY)} and (Fo, So) 9 (F, S). (See Figures 3-3a and 3-3b for a particular example of the objects and setting considered during this proof) Let: o'po :={xE Y-0 : If g E Eo and g

Q:o

u, then g n po =0

Then, define: c 0 :=

U (.

+ G[Uk] is the decomposition of G[6'O] into its connected components, we will let E = Xo,p, U {C-, ... , Uk}. We will use the acyclic orientation O of G' obtained from the two conditions 1) OEroK = Oo and 2) 71, - - I, l are maximal in (E, oz). The p.a.o. 0 is now obtained from OE, and let F be associated to 0 and S be obtained from ' =1 U ... U UkWe now prove that (Fo, So) C (F, S). if G[& 0'] = Gla1] + -

Since (Fo, So) C (F, S), it is enough to find x E F and z E So such that (x. z) C (F, S), so this is precisely what we will do. To begin, we note that for i E [k], the restriction Oi := Ool, is a p.a.o. of Gi := G[x], so we will let i be the connected partition of Gi and Of' the acyclic orientation of G' associated to Oi; moreover, we note that po intersects every element of i minimal in (Is, 5z). Hence, let us select go C po so that for every i E [k], go intersects every element of i minimal in (Yi, i) in exactly one point and so that go n -i contains only minimal elements in Oi. Now, take any x c F and let:

z

= E

E

depth

(j) -

ej

S.

iE[k] iEeoni

We will make use of the technique of Proposition 3.2.20.ii to prove that (x, z) E

64

00: Eo = {{1, 15}, {2, 18}, {3, 6, 7, 16}, {4, 13, 17}, {5,9,12}, {8}, {1

0: E

11},{14}}

{{1, 15}, {2, 3, 6, 7, 8, 16, 18}, {4, 13, 17}, {5,9, 12}, {10, 11}, {14}}

15

po = {1, 2 ,3 ,16 } w Eo,p, = {{4, 13, 17}, {5, 9, 12}, {14}, {10, 11}} 0o = {4, 5, 9, 10, 11, 12, 13, 14, 17}

E= 01

{{1, 15}}, {1, 15}, 02

{{2,18}, {3,6,7,16}, {8}} {2, 3,6,7,8, 16, 18}

2 = =

(b)

(a)

Figure 3-3: An example to the proof of Claim i.2 in Theorem 3.2.25. (F, S), so for that we need to consider a point in S, which we select as:

S=s=LN

X1~

E

iE[k]

S

ej (E S.

jEi

For i E [k], if we consider a pi E JG,(O) with pi , a-, Proposition 3.2.23 gives us:

)

depthQ d

j Qi

>N (eoi k jai

--

k

ji

/

\jEeingo

1

Ii12

lid

Jai I

_

N

k.IcrI| 2

> 0. Hence, for a sufficiently small e > 0, x + E(z - s) E F, so for each y E (x, z) we can find x' E F and s' E S such that y E (X', z'). That implies (FO, So) ; (F, S). Claim i.3 Let both F, S and FO, So satisfy the conditions of Claim i.b). Then, (F, S) n

#

0 if and only if F = FO and S = So. Moreover, (F, S) is a face of YG and dimff ((F, S)) = IpI + dimG[I(01) (similarly for (F,So)). Let O E (0, 1) and consider the polytope P, = {x E R[n] : i[]xi = El) + (1 - cx)N} n [ZG, NA]. Every x E P, satisfies the inequalities E

(n

+

(Fo, So)

Xi =

(1 - a)N + oc(n + JEJ) and Eje, xi ;> oc(lu-|+ E(G[u])I) for all u- C [n], a- =' 0. Per Claim i.1 and Claim i.2, the set (F, s) n P, can be characterized by the condition that it contains all the points x E P, which, among those inequalities,

65

satisfy the and only the following equalities:

Z xi = (1 -

(3.2.3)

c)N + cx(n + El) and

iE[n]

(3.2.4)

Exi = OC(jj + E(G[u-)), iE9

for all U E JG[p](Olp), o

\$

0.

This observation proves the first statement. For the second statement, we assume without loss of generality that N > n+ El and select generic coefficients 0, E R+ with U E JG[p](Ojp)\{0}, such that:

B:,(ju +

IE(G[u])I)

=

N - (n + El).

-E JG[p](O1- )\{O}

The linear functional,

f:=

Ze iE[n]

+

E UEJG[O](Oji)\{0}

f.Ze,

(3.2.5)

jEU

satisfies that, for x E P,

f(x) ;> (1 - oc)N + cv(n + El) + c (N - (n + |El))

=

N.

By the proof of the first claim, this inequality is tight if and only if x C (F, S) n P, = (F, S) n Po. Moreover, since this minimum is independent of c, the linear functional f is minimized in [ZG, NA] exactly at (F, S). If N < n + IEl, we must select negative coefficients and consider instead the maximum of the linear functional in question, analogously. For the third statement, we simply note that an open ball in the affine space determined by all x E R "n satisfying Equalities 3.2.3-3.2.4 can be easily (but tediously) found inside (F, S). Hence, dim,,. ((F, S)) = II + dimG[ ](OI).

Definition 3.2.27. Let G = G([n], E) be a connected simple graph. Let X% = (XG, - 0 if i E 9} : F E F(AG)} is, as suggested by our choice of notation, equal to the totally positive part of the fan of the graphical arrangement of G[o], regarding here R' as a subspace of RHn1. Per Theorem 3.2.15, since F(AG[,]) is precisely the normal fan of ZG], and F(AG[])+ the totally positive part of this fan, we know that the relatively open cones of F(AG[,)) correspond to the p.a.o.'s of G[a]. From the description of the cells of ZG[,], the cone C+, is exactly the cone in F(AG[,])+ normal to the cell of ZG[Y] corresponding to 0. This establishes the correspondence between cells of XG and elements of XG, since we can go both ways in this discussion. Using this same lens to regard cells of XG, the correctness of Definition 3.2.27.1 now follows from the analogous verification done in Theorem 3.2.25, by a standard result on normal fans of polytopes, namely, the duality of face containment.

3.3

Two ideals for acyclic orientations.

Definition 3.3.1. Let G = G([n, E) be a simple graph.

67

1. For an orientation 0 of G and for every i E [n], let:

j E [n]}I

Outdeg(G,1O)

11ij E

nod(G,O)(i)

,

indeg(G,O)(i) := {(j, i) E O[E] []jE[

7I

I{e E O[E] : either e = (j, i) or e = {i, j}, j E [nG}|,

where we denote the respective associated vectors in R["I as indeg(0 0 and nod(c,o). 2. For a C [n] with a \$ 0, define 1 Let now, for every i E [n]:

=

inof(G,,) (i)

:=

0

:= 1[n].

if i C o-, otherwise,

I{{i,j} E E :

Outof(Ga)()

outdeg(G,O),

ei,e E RNI, further writing 1

E E :j E a}

f{{ij}

),

[n]\}

0

if i E a, otherwise,

and denote the respective associated vectors of R [n] as inof(G,,) and outof(0,0). Remark 3.3.2. During this section, we will follow the notation and definitions of Miller and Sturmfels {2005}, Chapters 1,4,5,6 and 8, in particular, those pertaining to labelled polytopal cell complexes. We refer the reader to this standard reference on the subject for further details. Some key conventions worth mentioning here are: 1. The letter k will denote an infinite field.

2. For a := (ai, a2 ,.

. . , a,)

E NHn, ma := (x

: i

C [n]) is the ideal of k[x1,.

..

, xn]

associated to a. Definition 3.3.3. Let G = G([n], E) be a connected simple graph. The ideal AG of acyclic orientations of G is the monomial ideal of k[x1,... ,xn] minimally generated as: AG :-

Xindei(GO)

J

: 0 is an acyclic orientation of G).

indeg(Go)(i)+l

iE[nI Definition 3.3.4. Let G = G([n], E) be a connected simple graph. The tree ideal TG of G is the monomial ideal of k[x1,..., x.] minimally generated as:

TG :=

Xoutof(G,a)1c

x

utof(G,)(0+1

:9 E

2 [nI\{o}

and G[3-1 is connected)

t 3iEoc

Definition 3.3.5. Given two vectors a, b E

68

N En]

with b -- a (bi < ai

for all i E [n]),

let a\b be the vector whose i-th coordinate is: ai+ 1 - bi 0

ai\bi ar~i =

1, if bi if bi = 0.

If I is a monomial ideal whose minimal generators all divide x", then the Alexander dual of I with respect to a is: Ial := fl{ma\b

Xb

is a minimal generator of I}.

Theorem 3.3.6. Let G = G([n], E) be a simple connected graph. Then, the ideals AG and TG of Definitions 3.3.3-3.3.4 are Alexander dual to each other with respect to dG + 1, So AdG+1] = TG and T [dG+l

=

AG-

Proof. It is enough to prove one of these two equalities, so we will prove that A[dG 11 - TG. Take some o E 2["l\{o} such that G[o] is connected and consider Utof(G,fJ)M1. We will the minimal generator of TG given by XOUtOf(G,)+ l = verify that x"Utf(G,g)1' E m(dG+I)\b for every minimal generator Xb of AG. Se(G,) HE[] i lect an acyclic orientation 0 of G and let xindg(G,o) the minimal generator of AG associated to 0. If we take m E u to be maximal in ([n], 0 m and i E a imply i = m, then ( dG(m) outof(GM)(m)

+

)

\

outdeg(GO)(m) + 1 < ING(m)\oi + 1

(indeg(GO)(M) + )

+ 1. Hence,

Xoutof(G,+1,

E

OtOf(GU)(m)+1) C KdG(m)+1)\(inde9(GO)(m)+1) (dG+1A(indeg(G,O)

1)

This proves that TG C A[dG+l] Now, consider a monomial x' V TG with 0 -< b (so bi > 0 for some i E [n]). Then, for every U E 2" \{0 } there exists i E - such that bi < outof(G,cr)(i) + 1, noting here that the condition on G[o] being connected can be dropped. Hence, consider a bijective labeling f : [n] -+ [n] of the vertices of G such that bf-1(j) < outof(G,f-Il{,)(f(i) + 1 for all i E [n]. If we let 0 be the acyclic orientation of G such that for every e = {i,j} E E, O(e) = (ij) if and only if f(i) < f(j), then for all i E [n], bf-1(j) < outof(G'f-(1,i])(@) + 1 = outdeg(GO)(f-(i)) + 1 =

(dG(f'(i) + 1) \ (indeg(G,O)(fr(i)) + I therefore TG A [dG+1I that xb O A [dG+l, GG

,

or xb

'

M(dG+1)\(indeg(G,O)+1).

Corollary 3.3.7. Let G = G([n], E) be a simple connected graph. Then: AG

{minof(Ga)+1

a

E 2 [n]\{0} and G[-] is connected},

69

This shows

is the irreducible decomposition of AG. Also:

TG =

{moutdeg(0,G)+1 : 0 is an acyclic orientation of G},

is the irreducible decomposition of TG. Definition 3.3.8. For a simple connected graph G = G([n], E), consider the polytopal complexes ZG, YG and XG, which respectively realize the abstract cell complexes _TG, G and X of Definitions 3.2.11, 3.2.18 and 3.2.27. We will let ZG = (ZGz (YG, fy) and XG = (XG, x) be the Nin] -labelled cell complexes with underlying YG polytopal complexes given by ZG, YG and XG, respectively, and face labelling functions Ze 1 , EX, defined according to: 1. ZG: For a face F of ZG correspondingto 0 E 2G:

[n].

fz(F)i = nod(G,O)(i) + 1, i

2. YG: (a) For a face F of YG correspondingto (u0,O) E

f(F)

-

fy(F~i

G:

+ 1 if i E u,

nod(G[a],o)() dG(i) + 2

(b) For a face F of YG corresponding to u-

G

otherwise.

C2

dG(i) + 2 0

{]\{[n],0} g 3G:

if i Co, otherwise.

3. XG: For a face F of XG corresponding to (,-,O)

Ec

:

( outdeg(G[U], 0 )(i + outof(G,,(i) + 1 fF.0 otherwise.

if

E,

Lemma 3.3.9. Let G = G([n], E) be a simple connected graph. Then, for any face F of ZG with vertices v 1,... , V, we have that: Xez(F) = LCM {x z(vi)}iE[k]

where LCM stands for "least common multiple". Proof. Let F be a face of ZG with corresponding p. a. o. 0 of G and connected partition Y. Every acyclic orientation of G that corresponds to a vertex of F is obtained by 1) selecting an acyclic orientation for each of the G[o] with u CE, and then by 2) combining those III acyclic orientations with 0[E] n V 2 . For a fixed vertex i c o with u c 5, it is possible to select an acyclic orientation of G[-] in which i

70

is maximal and then to extend this to an acyclic orientation of G that refines 0, so if vertex v, of F corresponds to one such orientation, then e,(vj)j = nod(G,o)(i) + 1. On the other hand, clearly Ez(vj)i 5 nod(G,o)(i) + 1 for all vertices vj of F. Hence, xez(F) = L CM{xfz (v)}Elk]

Corollary 3.3.10. Similarly, for G as in Lemma 3.3.9 and for any face F of YG with vertices v 1 ,... ,V, we have that: X y(F) = LCM {x 4 (vi)}iE[k] where LCM stands for "least common multiple". Proof. If F is a face of YG inside the simplex NA, then this is immediate. If F corresponds to some (u, 0), then this is a consequence of the proof of Lemma 3.3.9, since the vertices of F are all the N - ej with i E [n]\c-, and all the vertices of YG that correspond to acyclic orientations of G whose restrictions to G[u] refine 0 and in which all edges of G connecting u with [n]\u are directed out of u.

Proposition 3.3.11. Let G = G([nl, E) be a simple connected graph. The cellular free complex FyG supported on YG is a minimal free resolution of the artinianquotient k~xi, . . . , x.]/ (AG + mdG+2). Proof. Without loss of generality, we assume here that N > n + IEj. From standard results in topological combinatorics it is easy to see that for b E N1"I, the closed b} form a faces of YG that are contained in the closed cone C-s = {v E R in) :v contractible polytopal complex, whenever this cone contains at least one face of YG. Now, suppose that b satisfies that bi 5 dG(i) + 1 for all i E [n]. Then, the complex of faces of YG in the cone Csb coincides with YG,- dG(i) + 2, and let Db = [n]\Ub. Consider the vector a e R["] such that: -

N

-{

ai= bi

if i E Ub,

if i E Db.

Then, the set of faces of YG in the cone C- 1 of the recursion to define fb(T), N < n, the edge that will be added to the set E (Tb(T)) is an edge of T. Initially, when N = 0, we have fb(T) (0) = p-1(0) = r and E (Tb(T)) = 0. Suppose that the result is true for all N < i, i E [n], and let us consider the i-th step of the recursion, so that K = {0,1,..., i - 1}. By induction, if j E [n]\fb(T)[K] and since 1 fb(T)(k) = p- (k) for all k E K, we have that {lo

so when

3, and select an arbitrary bijecttve-map fo : V -+ [n] (regarded as a labelling of V). Let us consider a sequence (ft)t=o,1,2,... of bijective maps V -+ [n] such that for t > 1, ft is obtained from ft-1 through the following random process: Let vt E V be chosen uniformly at random, and let, n if v = vt,

ft (v)=

ft i(v) - 1

if ft i(v) > ft i(v),

ft-i(v)

otherwise.

Consider now the sequence of acyclic orientations (Ot)t=o.1,2,... of G induced by the labellings (ft)t=0,1,2,..., so that for all e = {u, v} E E and t > 0, we have that Ot(e) = (u,v) if and only if ft(u) < ft(v). The sequence (Ot)t=o,1,2... is called the Cardshuffling (CS) Markov chain on the set of acyclic orientations of G. Equivalently, we can define this Markov chain by selecting an arbitrary acyclic orientation Oo of G, and then for each t > 1, letting Ot be obtained from Ot_1 by selecting vt e V uniformly at random and taking, for all e c E: Ot(e) =

Ot-i(e)

(v,vt)

if vt g e,

if e= {v,vt}.

Theorem 3.5.4. The Card-Shuffling Markov chain of G in Definition 3.5.3 is a welldefined, irreducible and aperiodic Markov chain on state space equal to the set of all acyclic orientations of G; its unique stationary distribution 7rC is given by: S=

,e(O) for all acyclic orientations 0 of G,

where e(O) denotes the number of linear extensions of the induced poset (V, 0 and Pr[Ot = Ot-1|Ot-1] > 0 for all t > 1), and has a unique stationary distribution 7rCs, necessarily then given by for every acyclic orientation 0 of G, from Equations 3.5.1. i(! 710 = Definition 3.5.5 (Edge-Label-Reversal Stochastic Process). Let G = G(V, E) be a connected simple graph with |VI = n, and select an arbitrary bijective map fo : V -+ [n] (regarded as a labelling of V). Let us consider a sequence (ft)t=0,1,2,... of bijective maps V -+ [n] such that for t > 1, ft is obtained from ft-1 through the following random process: Let et = {ut, vt} E E be chosen uniformly at random from this set, and let, if V = vt, ifv=ut,

ft-i(ut)

ft_1(Vt) ft_1(v)

ft(v)=

otherwise.

Consider now the sequence of acyclic orientations (0t)t=0,1,2,... of G induced by the labellings (ft)t=0,1,2,..., so that for all e = {u, v} E E and t > 0, we have that Ot(e) = (u, v) if and only if ft(u) < ft(v). The sequence (Ot)t=,1,2,... is called the Edge-LabelReversal (ELR) stochastic process on the set of acyclic orientations of G. Theorem 3.5.6. The Edge-Label-Reversal stochastic process of G in Definition 3.5.5 satisfies that, for every acyclic orientation 0 of G: N (irELR)

=ELR

:.

Jim

Lt

_L

=0]-

e(Q)

t=1

where e(O) denotes the number of linear extensions of the induced poset (V, this result holds independently of the initial choice of 00.

O), and

Proof. Consider the simple graph H on vertex-set equal to the set of all bijective maps V -+ [n], and where two maps f and g are connected by an edge if and only if there exists {u, v} E E such that f(u) = g(v), f(v) = g(u), and f(w) = g(w) for all w E V\{u, v}. Since G is connected, a standard result in the algebraic theory of the symmetric group shows that H is connected, e.g. consider a spanning tree T of G; then, any permutation in 6v can be written as a product of transpositions of the form (u v) with {u, v} E E(T). Moreover, by considering the parity of permutations in 6V, we observe that H is bipartite. Now, the sequence (ft)t=,1,2,... of Definition 3.5.5 87

is precisely a simple random walk on H, and the degree of each bijective map f : V -* [n] in H is clearly JE|, so the stationary distribution for this Markov chain is uniform. Necessarily then, the result follows from the construction of (Ot)t=o,1,2,... and Equations 3.5.1.

Definition 3.5.7 (Sliding-(n + 1) Stochastic Process). Let G = G(V, E) be a connected simple graph with|V| = n, and consider the graphGr. Let us select an arbitrary bijective map fo : V LI {r} -+ [n + 1], which we regard as a labelling of the vertices of Gr, and define a sequence (ft)t=o,1,2... of bijective maps V Li {r} -* [n + 1] such that for t > 1, ft is obtained from ft-1 through the following random process: Let Vt_1 E V U {r} be such that ft_1(vt-_) = n + 1, and select vt G NGr(vt-1) uniformly at random from this set. Then,

ft(v)=

n n+1

ifv=vt,

ft-1(vt) ft_1(v)

if V t=v_, otherwise.

Consider now the sequence of acyclic orientations (Ot)t=0,1,2,... of Gr induced by the labellings (ft)t=0,1,2,.., so that for all e = {u, v} E E(Gr) and t > 0, we have that Ot(e) = (u, v) if and only if ft(u) < ft(v). The sequence (Ot)t=0,1,2,... is called the Sliding-(n + 1) (SL) stochastic process on the set of acyclic orientations of Gr. Theorem 3.5.8. The Sliding-(n + 1) stochastic process of Gr of Definition 3.5.7 satisfies that, if Sr is the set of all acyclic orientations of Gr whose unique maximal element is r, then Z0 Pr[Ot Sr3= oc and for every 0 E Sr:

(nSL )O = 7TSL:0

liM

N-+o

LN

Nt= Nt=

Pr[Ot =0]

Pr[ Ot

Sr ]

e(O1)

n!

where Ov is the restriction of 0 to E (hence an acyclic orientation of G) and e(Ov) denotes the number of linear extensions of the induced poset (V, oi ). These results hold independently of the initial choice of 00. Proof. Consider the simple graph H on vertex-set equal to the set of all bijective maps V U {r} -+ [n + 1], and where two maps f and g are connected by an edge if and only if there exists {u, v} E E(Gr) such that f(u) = g(v) = n + 1, f(v) - g(u),

and f(w) = g(w) for all w E V\{u, v}. If two bijective maps

f, g

: V U {r}

-+

[n + 1]

differ only in one edge of Gr, so that f(u) = g(v) 7 n + 1 and f(v) = g(u) \$ n + 1 for some {u,.v} c E(Gr), but f(w) = g(w) for all w E V\{u, v}, then we can easily but somewhat tediously show that f and g belong to the same connected component of H, making use of the facts that vertex r is adjacent to all other vertices of Gr and that G is connected. But then, the proof of Theorem 3.5.6 shows that H is a connected graph. Now, the sequence (ft)t=o,1,2,... of Definition 3.5.7 is a simple random walk on H, and the degree of a bijective map f : V -+ [n] in H is clearly dc,(vi), where vf c V depends on f and is such that f(vf) = n + 1, so the stationary distribution

88

7r for this Markov chain satisfies that 7Tf = c - dG (vf), for some fixed normalization constant c E R+. The vertices of H that induce acyclic orientations of Gr from the set Sr are exactly the bijective maps f : V Li {r} -+ [n + 1] such that f(r) = n + 1, and for these we have that 7Tf = c - n. The result then follows from the construction of (Ot)t=0,1,2,... and from Equations 3.5.1.

Cov(0t_ 1 ) :=

{e E Ot-i[E] : e represents a cover relation in (V,

os)}

,

Definition 3.5.9 (Cover-Reversal Random Walk). Let G = G(V, E) be a simple graph with |Vj = n, and select an arbitrary acyclic orientation Oo of.G. Let us consider a sequence (Ot)t=0,1,2,... of acyclic orientations of G such that for t > 1, Ot is obtained from Ot_1 through the following random process: Let (u, v) be selected uniformly at random from the set,

and for all e E E, let,

if e = {u, v},

(v,u)

Ot(e) =

Oti(e) otherwzise.

The sequence (Ot)t=,1,2,... is called the Cover-Reversal (CR) random walk on the set of acyclic orientations of G. Theorem 3.5.10. The Cover-Reversal random walk in G of Definition 3.5.9 is a simple 2-period random walk on the 1-skeleton of the clean graphical zonotope ZG of Theorem 3.2.15 (hence, on a particularsimple connected bipartite graph on vertex-set equal to the set of all acyclic orientations of G), and its stationary distribution 7rcR satisfies that, for every acyclic orientation 0 of G:

7r

R =

c - ICOv(0),

where c E R+ is a normalization constant independent of 0. Proof. From the proof of Theorem 3.2.15, the edges 'of ZG are in bijection with the set of all p.a.o.'s 0 of G such that if 2 is the connected partition associated to 0, then II = n - 1. Hence, the edges of ZG are in bijection with the set of all pairs of the form (e, 0), where e E E and 0 is an acyclic orientation of the graph G/e, obtained from G by contraction of the edge e. The two vertices of ZG adjacent to an edge corresponding to a (e, 0) with e = {u, v} are, respectively, obtained from the acyclic orientations 01 and 02 of G such that 0 1 (e) = (u, v), 0 2 (e) = (v, u), and such that 021E\, are naturally induced by 0 (e.g. see Definition 3.2.4). Necessarily then, both (u, v) and (v, u) correspond respectively to cover relations in the posets (V, o,) and (V, 02), since otherwise the orientation 0 of G/e would not be acyclic. On the other hand, given an acyclic orientation 01 of G and an edge (u, v) E 01 [E] such that v covers u in (V, Oi), then, reversing the orientation of (only) that edge in 01 yields a new acyclic orientation 02 of G, so (v, u) E 02 [E]. Otherwise, using a directed-cycle formed by edges from 02 [E], which must then include the edge (v, u), 011E\e =

89

we observe that the relation u 1, Ot is obtained from Ot_1 through the following random process: Let {u, v} c E be selected uniformly at random from this set, with (u, v) E Ot_1[E], and for all e = {x, y} c E with (x, y) G Ot 1 [E], let, (e) OtOt e) -

{

,'X) (xY)

=

Ot- (e)

if u o, X