Extremal Graph Theory for Degree Sequences

Extremal Graph Theory for Degree Sequences∗ Xiao-Dong Zhang

arXiv:1510.01903v1 [math.CO] 7 Oct 2015

Department of Mathematics and MOE-LSC Shanghai Jiao Tong University 800 Dongchuan road, Shanghai, 200240, P.R. China Email: [email protected]

Abstract This paper surveys some recent results and progress on the extremal problems in a given set consisting of all simple connected graphs with the same graphic degree sequence. In particular, we study and characterize the extremal graphs having the maximum (or minimum) values of graph invariants such as (Laplacian, p-Laplacian, signless Laplacian) spectral radius, the first Dirichlet eigenvalue, the Wiener index, the Harary index, the number of subtrees and the chromatic number etc, in given sets with the same tree, unicyclic, graphic degree sequences. Moreover, some conjectures are included.

Key words: Graphic degree sequence, majorization, tree, unicyclic graphs, spectral radius, Wiener index. AMS Classifications: 05C50, 05C05, 05C07, 05C12, 05C35

1

Introduction

A nonincreasing sequence of nonnegative integers π = (d0 , d1, · · · , dn−1 ) with d0 ≥ d1 ≥ · · · ≥ dn−1 is called graphic if there exists a simple graph having π as its vertex degree sequence and such a graph is called a realization of π. Erdös and Gallai [17] characterized all graphic degree sequences. Supported by National Natural Science Foundation of China (No.11271256) and Innovation Program of Shanghai Municipal Education Commission (No:14ZZ016) ∗

1

Theorem 1.1 [17] A nonincreasing sequence of nonnegative integers π = (d0 , d1, · · · , Pn−1 dn−1 ) is graphic degree sequence if and only if i=0 di is even and r X

di ≤ r(r + 1) +

i=0

n−1 X

min{r + 1, di },

i=r+1

for r = 0, 1, · · · , n − 2. Moreover, Senior [41] and Hakimi[24] characterized all graphic degree sequences with at least a realization being connected. Theorem 1.2 ([41],[24]) A nonincreasing sequence of nonnegative integers π = (d0 , d1 , · · · , dn−1) is graphic sequence with at least a realization being connected if and only if Pn−1 i=0 di is even and r X

di ≤ r(r + 1) +

Pn−1 i=0

min{r, di },

i=r+1

i=0

for r = 0, 1, · · · , n − 2,

n−1 X

di ≥

n(n−1) 2

and dn−1 ≥ 1.

Let π be graphic degree sequence with at least a realization being connected. Denote by G(π) the set of all connected graphs which are realizations of π, i.e., G(π) = {G |G is connected and has π as its degree sequence}.

(1)

Sierksma and Hoogeveen [42] surveyed seven criteria for a nonnegative integer sequence being graphic. For graphic sequences, Ferrara [19] surveyed recent research progress and new results on graphic sequences and presented a number of approachable open problems. Extremal graph theory is an extremal important branch of graph theory. One is interested in relations among the various graph invariants, such as order, size, connectivity, chromatic number, diameter and eigenvalues, and also in the values of these invariants which ensure that the graph has certain properties. Bollobás published an excellent book [7] and recent survey in [8]. Erdös, Jackson and Lehel in [18] proposed the problem on the possible clique number attained by graphs with the same degree sequence, i.e., determine the smallest integer number M so that each graphic sequence π = (d0 , · · · , dn−1 ) with d0 ≥ d1 ≥ · · · ≥ dn−1 ≥ 1 and Pn−1 i=0 di ≥ M has a realization G having given clique number. Li and Yin [32] 2

surveyed extremal graph theory and degree sequences, in particular for the above problem and its variants. On the other hand, many graphs coming from the real world have power law degree distribution: the number of vertices with degree k is proportional for k −β for some exponent β ≥ 1. Chung and Lu (for example, see [11], [12]) proposed the famous Chung-Lu’s model: random graphs with given expected degree sequences and studied their properties such as diameters, eigenvalues, the average distance, etc. In this survey, we just focus on some extremal properties of graphs with given degree sequences. Graph invariants such as the spectral radius, the Dirichlet eigenvalues, the Wiener index, the Harary index, the number of subtrees, etc., will be considered. Graphic degree sequences such as tree degree sequences, unicyclic degree sequences etc, will be involved. Moreover, the majorization theorem on two different degree sequences are obtained. Therefore, we may propose the following general problem. Problem 1.3 For a given graphic degree sequence π, let Gπ = {G | G is connected with π as its degree sequence}. For some graph invariants, such as spectral radius, the Wiener index, etc, find the maximum (minimum) value of these graph invariants in Gπ and characterize all extremal graphs which attain these values.

2

Preliminary

In this section, we introduce some notions and properties. For a graph G = (V, E) with a root v0 , denote by dist(v, v0 ) the distance between vertices v and v0 . Moreover, the distance dist(v, v0 ) is called the height h(v) = dist(v, v0 ) of a vertex v. Definition 2.1 ([4], [61]) Let G = (V, E) be a graph with root v0 . A well-ordering ≺ of the vertices is called a breadth-first search ordering (BFS-ordering for short) if the following holds for all vertices u, v ∈ V : (1) u ≺ v implies h(u) ≤ h(v); (2) u ≺ v implies d(u) ≥ d(v); (3) let uv ∈ E(G), xy ∈ E(G), uy ∈ / E(G), xv ∈ / E(G) with h(u) = h(x) = h(v) − 1 = h(y) − 1. If u ≺ x, then v ≺ y. We call a graph that has a BFS-ordering of its vertices a BFS-graph. 3

For example, for a given degree sequence π = (4, 4, 3, 3, 3, 3, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1), is the tree of order 17 (see Fig.1). There is a vertex v01 in layer 0; four vertices v11 , v12 , v13 , v14 in layer 1; nine vertices v21 , v22 , · · · , v29 in layer 2; three vertices v31 , v32 , v33 in layer 3. Tπ∗

❝ ✁ ❆ v21 ✁ ❆ ❆❝ ❝✁

v31

❝ v01 ✏✏ P ✏ ❅PPP ✏ ✏ PP ✏ ❅ PP ✏✏ ✏ PP ❅ ✏ ✏ PP ✏ ❅ PP ✏✏ ❅❝ P❝ v ❝✏ ❝ v11 v12 v13 14 ❅ ❅ ❅ ❅ ❅ ❅ ❅ ❅ ❅ ❅ ❅ ❅ ❅❝ ❝ ❅❝ ❅❝ ❝ ❅❝ ❝ ❝

v22

v23 v24

v25 v26

v27 v28

v29

❝

v32 v33 Figure 1

It is easy to see that every tree has an ordering such that it satisfies the conditions (1) and (3) by using breadth-first search, but not every tree has a BFS-ordering. For example, the following tree T of order 17 does not have a BFS-ordering with degree sequence π = (4, 4, 3, 3, 3, 3, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1) (see Fig.2).

❝

v22

❝ v01 ✏✏ P ✏ ❅PPP ✏✏ PP ✏ ❅ ✏ PP ✏ PP ❅ ✏✏ ✏ PP ✏ ❅ ✏ PP ✏ ❅ ✏ P❝ v ❝ v ❝v ❝v 11 12 13 14 ❅ ❅ ❅ ❅ ❅ ❅ ❅ ❅ ❅ ❅ ❅ ❅ ❅❝ ❝ ❅❝ ❅❝ ❝ ❅❝ ❝ ❝ v29 v23 v24✁ ❆ v21 v25 v26 v27 v28 ✁ ❆ ❆❝ ❝✁ ❝

v33

v31

v32

Figure 2

Moreover, it is easy to see [4] that there may be many BFS graphs for a given graphic degree sequence. Another importation notion is majorization. Let x = 4

(x1 , x2 , · · · , xp ) and y = (y1 , y2 , · · · , yp ) be two nonnegative integers. We arrange the entries of x and y in nonincreasing order x↓ = (x[1] , · · · x[p] ) and y↓ = (y[1] , · · · , y[p]). If k X i=1

x[i] ≥

k X

y[i] , for k = 1, · · · , p,

(2)

i=1

x is said to weakly majorize y and denoted x ⊲w y or y ⊳w x. Further, if y ⊳w x and p X

x[i] =

p X

y[i],

(3)

i=1

i=1

x is said to majorize y and denoted by x ⊲ y or y ⊳ x. For details, the readers are referred to the book of Marshall and Olkin [35]. It is well known that the following result holds (see [17] or [53]). Proposition 2.2 ([17],[53]) Let π = (d0 , · · · dn−1 ) and π ′ = (d′0 , · · · , d′n−1) be two different nonincreasing graphic degree sequences. If π ⊳ π ′ , then there exists a series graphic degree sequences π1 , · · · , πk such that π ⊳ π1 ⊳ · · · ⊳ πk ⊳ π ′ , and only two components of πi and πi+1 are different from 1. Let G = (V, E) be a simple graph with vertex set V (G) = {v1 , · · · , vn } and edge set E(G). There are many matrices associated with a graph. Let A(G) = (aij ) be the (0, 1) adjacency matrix of G, where aij = 1 if vi and vj are adjacent, and 0 elsewhere. The spectral radius of A(G) is denoted by ρ(G) and called the spectral radius of G. Moreover, let f be the unit eigenvector of A(G) corresponding to ρ(G). By the Perron-Frobenius theorem, f is unique and positive. So f is called Perron vector of G. In addition, denote by d(vi ) the degree of vertex vi and D(G) = diag(d(u), u ∈ V ) the diagonal matrix of vertex degrees of G, thus the matrix L(G) = D(G) − A(G) is called the Laplacian matrix of a graph G. The spectral radius of L(G) is equal to the largest eigenvalue of L(G) and denoted by λ(G). Moreover, λ(G) is called the Laplacian spectral radius of G. L(G) = D −1/2 L(G)D −1/2 is call the normal Laplacian matrix of a graph. The spectral radius of L(G) is denoted by µ(G) and called the normal Laplacian spectral radius. In addition, the matrix Q(G) = D(G) + A(G) is called the signless Laplacian matrix of G. The spectral radius of Q(G) is denoted by q(G) and called the signless Laplacian spectra radius. Let g be the unit eigenvector of Q(G) corresponding to eigenvalue q(G) and be called signless Laplacian Perron vector of G. The adjacency, Laplacian, normal Laplacian and signless Laplacian matrices play a key role in spectral graph theory. The book “ Spectra of graph” 5

[13] by Cvetković, Doob abd Sachs focuses on the adjacency matrix. The book “ Applications of combinatorial matrix theory to Laplacian matrices of graphs” [37] by Molitierno studies many properties of Laplacian matrices. The book “ Spectral graph theory” [10] by Chung focuses on normal Laplacian matrix. Moreover, there are several surveys on the topic of signless Lapacian matrices (see [14] and the references therein).

3

Eigenvalues with given tree degree sequences

Pn−1 If a nonnegative sequence π = (d0 , · · · , dn−1 ) is graphic and i=0 di = 2(n − 1), π is called tree degree sequence and denoted by Tπ the set of all trees having degree sequence π. For a tree degree sequence π = (d0 , d1 , · · · , dn−1 ) with d0 ≥ d1 ≥ · · · ≥ dn−1 , With the aid of breadth-first search method, we can define a special tree Tπ∗ with degree sequence π as follows. Assume that dm > 1 and dm+1 = · · · = dn−1 = 1 for 0 ≤ m < n − 1. Put s0 = 0. Select a vertex v01 as a root and begin with v01 in layer 0. Put s1 = d0 and select s1 vertices {v11 , · · · , v1,s1 } in layer 1 such that they are adjacent to v01 . Thus d(v01 ) = s1 = d0 . We continue to construct all other layers by recursion. In general, put st = ds0 +s1 +···+st−2 +1 + · · · + ds0 +s1 +···+st−2 +st−1 − st−1 for t ≥ 2 and assume that all vertices in layer t have been constructed and are denoted by {vt1 , · · · , vtst } with d(vt−1,1 ) = ds0 +···+st−2 +1 , · · · , d(vt−1,st−1 ) = ds0 +···+st−1 . Now using the induction hypothesis, we construct all vertices in layer t + 1. Put st+1 = ds0 +···+st−1+1 +· · ·+ds0 +···+st −st . Select st+1 vertices {vt+1,1 , · · · vt+1,st+1 } in layer t+1 such that vt+1,i is adjacent to vtr for r = 1 and 1 ≤ i ≤ ds0 +···+st−1 +1 −1 and for 2 ≤ r ≤ st and ds0 +···+st−1 +1 +ds0 +···+st−1 +2 +· · ·+ds0 +···st−1 +r−1 −r+2 ≤ i ≤ ds0 +···+st−1+1 + ds0 +···+st−1+2 + · · · + ds0 +···+st−1 +r − r. Thus d(vtr ) = ds0 +···+st−1+r for 1 ≤ r ≤ st . Assume that m = s0 +· · ·+sp−1 +q. Put sp+1 = ds0 +···+sp−1 +1 +· · ·+ds0 +···+sp−1 +q −q and select sp+1 vertices {vp+1,1, · · · , vp+1,sp+1 } in layer p + 1 such that vp+1,i is adjacent to vpr for 1 ≤ r ≤ q and ds0 +···+st−1 +1 +· · ·+ds0 +···+sp−1 +2 +· · ·+ds0 +···sp−1 +r−1 −r+2 ≤ i ≤ ds0 +···+sp−1 +1 + · · · + ds0 +···+sp−1 +2 + · · · + ds0 +···+sp−1 +r − r. Thus d(vp,i ) = ds0 +···+sp−1 +i for 1 ≤ i ≤ q. In this way, we obtain a tree Tπ∗ . It is easy to see that Tπ∗ has degree sequence π. Moreover, this special tree Tπ∗ is called greedy tree with a given tree degree sequence π. In addition, it is easy to show the following assertion holds. Proposition 3.1 ([4], [60]) Given a tree degree sequence π, there exists a unique 6

tree Tπ∗ with degree sequence π having a BFS-ordering. In other words, the greedy tree with π is only tree having a BFS-ordering in the set Tπ . Moreover, any two trees with the same degree sequences and having BFS-ordering are isomorphic.

3.1

The (Laplacian, signless Laplacian) spectral radius

Biyiko˘glu and Leydold [4]characterized all extremal trees having the largest spectral radius in the set Tπ consisting of all trees with a given degree sequence π. While, Zhang [60] characterized all extremal trees having the largest (signless) Laplacian spectral radius in the set Tπ consisting of all trees with a given degree sequence π, since the signless Laplacian radius of a tree T is always equal to its the Laplacian spectral radius. It is not a surprise that the two extremal graphs are coincident. Theorem 3.2 ([4], [60]) For a given tree degree sequence π, the greedy tree Tπ∗ is the only tree having the largest spectral radius (resp. signless Laplacian radius) in Tπ . Moreover, The BFS-ordering of the greedy tree is consistent with the Perron vector f of G in such a way that f (u) > f (v) (resp. g(u) > g(v)) implies u ≺ v. Theorem 3.3 ([4], [60]) Let π and τ be two different tree degree sequences with the same order. Let Tπ∗ and Tτ∗ have the largest spectral radii (resp. Laplacian, signless Laplacian spectral radii) in Tπ and Tτ , respectively. If π ⊳ τ , then ρ(Tπ∗ ) < ρ(Tτ∗ ), λ(Tπ∗ ) < λ(Tτ∗ ) and q(Tπ∗ ) < q(Tτ∗ ). With the aid of Theorems 3.2 and 3.3, we may deduce extremal graphs with the largest (resp. signless Laplacian) spectral radius in some class of graphs. (1)

Corollary 3.4 ([4],[60]) Let Tn,s be the set of all trees of order n with s leaves and let Tπ∗ be the greedy tree with π = (t, 2, · · · , 2, 1, · · · , 1) (i.e., if n − 1 = sq + t, 0 ≤ t < s, then Tπ∗ is obtained from t paths of order q + 2 and s − t paths of order q + 1 by identifying one end of the s paths.). Then ρ(T ) ≤ ρ(Tπ∗ ) (resp. λ(T ) ≤ λ(Tπ∗ ), (1) q(T ) ≤ q(Tπ∗ )) for any tree T ∈ Tn,s , with equality if and only if T = Tπ∗ . (2)

Corollary 3.5 ([60]) Let Tn,∆ be the set of all trees of order n with the maximum degree ∆ ≥ 3 and let Tπ∗ be the greedy tree with π which is defined as follows: Denote p −2 ⌉ − 1 and n − ∆(∆−1) = (∆ − 1)r + q for 0 ≤ q < ∆ − 1. If p = ⌈log(∆−1) n(∆−2)+2 ∆ ∆−2 p−1 −2 q = 0, put π = (∆, · · · , ∆, 1, · · · , 1) with the number ∆(∆−1) + r of degree ∆. If ∆−2 ∆(∆−1)p−1 −2 1 ≤ q, put π = (∆, · · · , ∆, q, 1, · · · , 1) with the number + r of degree ∆. ∆−2 (2) Then ρ(T ) ≤ ρ(Tπ∗ ) (resp. λ(T ) ≤ λ(Tπ∗ ) , q(T ) ≤ q(Tπ∗ )) for any tree T ∈ Tn,∆ , with equality if and only if T = Tπ∗ . 7

(3)

Corollary 3.6 ([59],[60]) Let Tn,α be the set of all trees of order n with the independence number α and let Tπ∗ be the greedy tree with π = (α, 2, · · · , 2, 1, · · · , 1) the numbers n − α − 1 of 2 and α of 1. Then ρ(T ) ≤ ρ(Tπ∗ ) (resp. λ(T ) ≤ λ(Tπ∗ ) , (3) q(T ) ≤ q(Tπ∗ )) for any tree T ∈ Tn,α , with equality if and only if T = Tπ∗ . (4)

Corollary 3.7 ([22], [60]) Let Tn,β be the set of all trees of order n with the matching number β and Tπ∗ be the greedy tree with π = (n − β, 2, · · · , 2, 1, · · · , 1) and the number n − β of 1. Then ρ(T ) ≤ ρ(Tπ∗ ) (resp. λ(T ) ≤ λ(Tπ∗ ) , q(T ) ≤ q(Tπ∗ )) for any tree (4) T ∈ Tn,β , with equality if and only if T = Tπ∗ . On the minimum (Laplacian, signless Laplacian) spectral radius of with a given tree degree sequence, it seems to be more difficult. The related results are referred to [6].

3.2

The spectral radius of p-Laplacian of weighted trees

Now we turn to consider p-Laplacian eigenvalues of weighted graphs. Let GW = (V (G), E(G), W (G)) be a weighted graph with vertex set V (G) = {v0 , v1 , · · · , vn−1 }, edge set E(G) and weight set W (G) = {wk > 0, k = 1, 2, · · · , |E(G)|}. If uv ∈ E(G), denote by wG (uv) the weight of an edge uv; if uv ∈ / E(G), define wG (uv) = 0. The weight of a vertex u, denoted by wG (u), is the sum of weights of all edges incident to u in G. For p > 1, the discrete p-Laplacian △p (G) of a function f on V (G) is defined to be X △p (G)f (u) = (f (u) − f (v))[p−1]wG (uv), (4) v,uv∈E(G)

where x[q] = sign(x)|x|q . When p = 2, △2 (G) is the well-known graph Laplacian (see [5]), i.e., ∆2 (G) = L(G) = D(G) − A(G), where A(G) = (wG (vi vj ))n×n denotes the weighted adjacency matrix of G and D(G) = diag(wG (v0 ), wG (v1 ), · · · , wG (vn−1 )) denotes the weighted diagonal matrix of G. A real number λ is called an eigenvalue of △p (G) if there exists a function f 6= 0 on V (G) such that for u ∈ V (G), ∆p (G)f (u) = λf (u)[p−1].

(5)

The function f is called the eigenfunction corresponding to λ. The largest eigenvalue of ∆p (G), denoted by λp (G), is called the p-Laplacian spectral radius. 8

Given a tree degree sequence π and a positive set W , denote by Tπ,W the set of trees with a given tree degree sequence π and a positive weight set W . Thus the un-weighted greedy tree Tπ∗ has a BFS-ordering v0 ≺ v2 · · · ≺ vn−1 . Hence the unique ∗ weighted greedy tree Tπ,W is obtained from the un-weighted greedy tree Tπ∗ whose edge weighted has the following property“ for any two edges vi vj and vk vl with i < j, k < l, weight w(vi vj ) ≥ w(vk vl ) if i < k or i = k and j < l.” Zhang and Zhang [55] proved the following results Theorem 3.8 [55] For a given tree degree sequence π and a positive weight set W , ∗ the weighted greedy tree Tπ,W is the unique weighted tree with the largest p-Laplacian spectral radius in Tπ,W , which is independent of p. ∗ Theorem 3.9 [55] Let π and τ be two different tree degree sequences. Let Tπ,W ∗ and Tτ,W be two weighted greedy trees in Tπ,W and Tτ,W respectively. If π ⊳ τ , then p ∗ ∗ λ (Tπ,W ) < λp (Tτ,W ).

Remark If the positive set W consisting of all 1, Theorems 3.8 and 3.9 become Biyiko˘glu and Leydold’s result [5]. If p = 2, Theorems 3.8 and 3.9 become Tan’s result [48]. If the positive set W consisting of all 1 and p = 2, Theorems 3.8 and 3.9 become Zhang’ result[60]. In addition, Corollaries 3.4, 3.5, 3.6 and 3.7 can be easily generalized to the weighted trees, respectively. Moreover, the p-adjacency eigenvalue may be similarly defined and Theorems 3.8 and 3.9 can be translated to the p-adjacency eigenvalue.

3.3

The Dirichlet eigenvalue of trees with boundary

In this subsection, we consider the Dirichlet eigenvalues of trees with boundary. A graph with boundary G = (V0 ∪∂V , E0 ∪∂E) consists of interior vertex set V0 , boundary vertex set ∂V , interior edge set E0 that connect interior vertices, and boundary edge set ∂E that join interior vertices with boundary vertices (for example, see [10] or [21]). Throughout this subsection, we always assume that the degree of any boundary vertex is 1 and the degree of any interior vertex is at least 2. A real number λD is called a Dirichlet eigenvalue of G if there exists a function f 6= 0 such that they satisfy the Dirichlet eigenvalue problem: ( L(G)f (u) = λf (u) u ∈ V0 ; f (u) = 0 u ∈ ∂V. 9

The function f is called an eigenfunction corresponding to λD . Recently, Bıyıko˘glu and Leydold [3] proposed the following problem: Problem 3.10 ([3]) Give a characterization of all graphs in a given class C with the Faber-Krahn property, i.e., characterize those graphs in C which have minimal first Dirichlet eigenvalue for a given “volume”. In order to study the above problem, Bıyıko˘glu and Leydold [3] extended the concept of an SLO*-ordering for describing the trees with the Faber-Krahn property, which was introduced by Pruss (see [38]). The notation of an SLO*-ordering may be extended for any connected graphs. Definition 3.11 ([3]) Let G = (V0 ∪ ∂V, E0 ∪ ∂E) be a connected graph with root v0 . Then a well-ordering ≺ of the vertices is called spiral-like (SLO*-ordering for short) if the following holds for all vertices u, v, x, y ∈ V0 ∪ ∂V : (1) v ≺ u implies h(v) ≤ h(u), where h(v) denotes the distance between v and v0 ; (2) Let uv ∈ E(G), xy ∈ E(G), uy ∈ / E(G), xv ∈ / E(G) with h(u) = h(v) − 1 and h(x) = h(y) − 1. If u ≺ x, then v ≺ y ; (3) If v ≺ u and v ∈ ∂V , then u ∈ ∂V . (4) If v ≺ u and v, u ∈ V0 , then d(v) ≤ d(u). Clearly, if G is a tree, an SLO*-ordering of G is consistent with the definition of an SLO*-ordering in [3]. Moreover, if there exists a positive integer r such that the number of vertices v with h(v) = i + 1 is not less than the number of vertices v with h(v) = i for i = 1, · · · , r − 1, and h(u) ∈ {r, r + 1} for any boundary vertex u ∈ ∂V , G is called a ball approximation. Given a tree degree sequence π, denote by Tπ,B the set consisting of all trees with boundary and degree sequence π. Bıyıko˘glu and Leydold [3] proved that Tπ,B contains an SLO*-tree that is uniquely determined up ∗ to isomorphism. This special tree in Tπ,B is denoted by Tπ,B . Theorem 3.12 [3] For a given tree degree sequence π, ∗ ) for T ∈ Tπ,B λD (T ) ≥ λD (Tπ,B ∗ with equality if and only if T = Tπ,B .

Theorem 3.13 [3] Let π and τ be two different tree degree sequences. If π ⊳ τ , then ∗ ∗ λD (Tπ,B ) > λD (Tτ,B ). 10

4

Chemical indices for tree degree sequences

There are many indices which are used to describe molecules and molecular compounds in chemical graph theory. One of the most widely known topological descriptor is the Wiener index which is named after chemist Wiener [54] who firstly considered it. The Wiener index of a graph G is defined as X W (G) = d(vi , vj ). (6) {vi ,vj }⊆V (G)

Let T = (V, E) be a rooted tree with root r. For each vertex u, let T (u) be the subtree of the rooted tree T induced by u and all its successors in T . In other words, if u is not the root r of tree T and v is the parent of u, then T (u) is the connected component of T obtained from T by deleting the edge uv such that the component does not contain the root r; if u is the root r, then T (u) is the tree T . Let φT (u) = |T (u)| be the number of vertices in T (u) and denote φ(T ) = (φT (u), u ∈ V (T )). We prove the following results Theorem 4.1 [64] Let T be a rooted tree in Tπ . Then the following conditions are equivalent: (1) T has a BFS-ordering; (2) φ(T )↓ = φ(Tπ∗ )↓ ; (3) T is isomorphic to Tπ∗ . Further, this result can be used to characterize the trees having the minimum Wiener index among all trees with a given degree sequence. Wang [52] independently proved the same result by a different approach. Theorem 4.2 ([52],[64]) Given a tree degree sequence π, the greedy tree Tπ∗ is a unique tree with the minimum Wiener index in Tπ . Theorem 4.3 [64] Let π and τ be two different tree degree sequences. Let Tπ∗ and Tτ∗ be two the greedy trees in Tπ and Tτ , respectively. If π ⊳ τ , then W (Tπ∗ ) < W (Tτ∗ ). On the other hand, it is natural to investigate the extremal trees which attain the maximum Wiener index among all trees with given degree sequences. But this problem seems to be difficult, since it is discovered that the extremal trees are not unique and depend on the values of components of degree sequences. However, there are some partial results. A caterpillar is a tree with the property that a path remains if all leaves are deleted and this path is called spine. 11

Theorem 4.4 [43] If a tree Tπ♯ has the maximum Wiener index among all trees in Tπ for a given tree degree sequence π, then Tπ♯ is a caterpillar. Theorem 4.5 [63] Let π = (d0 , · · · , dn−1 ) with d0 ≥ · · · ≥ dk−1 ≥ 2 ≥ dk+1 = · · · = dn−1 = 1. If a caterpillar Tπ♯ has the maximum Wiener index among all trees in Tπ and the spine is v0 v1 · · · vk−1 with d(v0 ) ≥ d(vk−1), then there exists a 1 ≤ t ≤ k − 2 such that d(v0 ) ≥ d(v1 ) ≥ · · · ≥ d(vt ) and d(vt ) ≤ d(vt+1 ) ≤ · · · ≤ d(vk−1 ). Further, Schmuck, Wagner and Wang [40] proposed and studied a new graph invariant which are based on distance. Theorem 4.6 [40] Let π = (d0 , d1 , · · · , dn−1 ) be a tree degree sequence let r be an arbitrary positive integer. If pr (T ) is the number of pairs (u, v) of vertices such that d(u, v) ≤ r for a tree T ∈ Tπ , then pr (T ) ≤ pr (Tπ∗ ) with equality for all r = 1, · · · , n−1 if and only if T is the greedy tree Tπ∗ . Recently, Schmuck, Wagner and Wang [40] proposed a general graph invariant Wψ (T ) =

X

ψ(d(u, v))

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{u,v}⊆V (G)

for any nonnegative function ψ. Further, they proved the following result. Theorem 4.7 [40][51] Let π be a tree degree sequence and ψ(x) be any nonnegative and nondecreasing (resp. nonincreasing) function on x. Then for any tree T ∈ Tπ , Wψ (T ) ≥ (resp. ≤) Wψ (Tπ∗ ).

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Further, if ψ(x) is strictly increasing function, then equality holds if and only if T is the greedy tree Tπ∗ . , Remark. If ψ(x) = x, then Wψ (T ) is the classical Wiener index. If ψ(x) = x(x+1) 2 then Wψ (T ) is the hyper-Wiener index [29]. If ψ(x) = x1 , then Wψ (T ) is the Harary index. Moreover, it is easy to deduce the following result from [40] and [51]. Theorem 4.8 Let π and τ be two different tree degree sequences with π ⊳ τ . If ψ(x) is any nonnegative and strictly increasing (resp. decreasing) function on x, then Wψ (Tπ∗ ) > (resp. Wψ (Tτ∗ ). Theorem 4.9 [40] Let π be a tree degree sequence and ψ(x) be strictly increasing and convex function on x. If Tπ♯ is a tree that maximizes Wψ (T ) among all trees in the set Tπ , then Tπ♯ is a caterpillar. In order to study total π-electron energy on molecular structure, Gutman and Trinajstić [23] proposed the second Zagreb index, which is defined to be X ZA(T ) = d(u)d(v), (10) uv∈E(T )

where d(u) and d(v) are the degrees of vertices u and v, respectively. Theorem 4.10 [40] Let π be a tree degree sequence. Then the greedy tree Tπ∗ is the only tree having the maximum second zagreb index among all trees in T ∈ Tπ , i.e., for any tree T ∈ Tπ , ZA(T ) ≤ ZA(Tπ∗ ) (11) with equality if and only if T is the greedy tree Tπ∗ . Let G be a simple graph. Denoted by m(G, k) the number of k−matchings in G and m(G, 0) = 1. The Hosoya index, named after Haruo Hosoya, is defined to be X m(G, k) (12) Z(G) = k≥0

Denote by i(G, k) the number of k−independent sets in G and i(G, 0) = 1. The Merrifield-Simmons index is defined to be X i(G, k) (13) MS(G) = k≥0

13

The energy of graph is defined by the sum of the absolute values of all the eigenvalues of A(G), and denoted by En(G). Andriantiana [1] proved that the greedy tree has minimum energy, Hosoya index and maximum Merrifield-Simmons index. Theorem 4.11 [1] Let π be a tree degree sequence. Then the greedy tree Tπ∗ is the only tree having minimum energy, minimum Hosoya index and maximum MerrifieldSimmons indices, respectively among all trees in T ∈ Tπ , i.e., for any tree T ∈ Tπ , Z(T ) ≥ Z(Tπ∗ ), En(T ) ≥ En(Tπ∗ ), MS(T ) ≤ MS(Tπ∗ ),

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with any one equality if and only if T is the greedy tree Tπ∗ . Theorem 4.12 [1] Let π and τ be two different tree degree sequences. Then the greedy trees Tπ∗ and Tτ∗ are the minimum energy, minimum Hosoya index and maximum Merrifield-Simmons indices in Tπ and Tτ respectively. If π ⊳ τ , then En(Tπ∗ ) > En(Tτ∗ ), Z(Tπ∗ ) > Z(Tτ∗ ), MS(Tπ∗ ) < MS(Tτ∗ ).

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In addition, there are the following results on the number of subtrees of a tree. Theorem 4.13 [65] With a given degree sequence π, Tπ∗ is the unique tree with the maximum number of subtrees in Tπ . Theorem 4.14 [65] Given two different degree sequences π and τ . If π ⊳ τ , then the number of subtrees of Tπ∗ is less than the number of subtrees of Tτ∗ .

5

Unicyclic degree sequences

For a given nonincreasing degree sequence π = (d0 , d1 , · · · , dn−1) of a unicyclic graph with n ≥ 3, we construct a special unicyclic graph Uπ∗ as follows: If d0 = 2, denote by Uπ∗ the cycle of order n. If d0 ≥ 3, denote by Uπ∗ is the graph obtained from a cycle C3 of order 3 and by breadth-first search ordering, i.e., Uπ∗ is BFS graph with the first three vertices, each pair of which is adjacent. Clearly, there exists only one such unicyclic graph Uπ∗ for a given unicyclic degree sequence π, which is called the greedy unicyclic graph with a given unicyclic degree sequence π. Proposition 5.1 ([24], [41]) Let π = (d0 , · · · , dn−1) be a positive nonincreasing inPn−1 di = 2n teger sequence with n ≥ 3. Then π is unicyclic graphic if and only if i=0 and d2 ≥ 2. 14

Theorem 5.2 ([33], [61]) Let π = (d0 , d1 , · · · , dn−1 ) be a unicyclic degree sequence. Then the greedy unicyclic graph Uπ∗ is the only unicyclic graph having the largest (signless Laplacian) spectral radius in Uπ , i.e., ρ(G) ≤ ρ(Uπ∗ ),

q(G) ≤ q(Uπ∗ )

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with one equality if and only if G is the greedy unicyclic graph Uπ∗ Theorem 5.3 ([33], [61]) Let π and τ be two different degree sequences of unicyclic graphs with the same order. If π ⊳ τ then ρ(Uπ∗ ) < ρ(Uτ∗ ) and q(Uπ∗ ) < q(Uτ∗ ) In order to present the results on the Dirichlet eigenvlaues, we need the following notion. Let π = (d0 , · · · , dk−1, 1, · · · , 1) be a unicyclic degree sequence with d0 ≥ d1 ≥ ∗ . . . ≥ dk−1 . If d2 ≥ 3 , then let Uπ,B have SLO*-ordering with each pair of the first three vertices being adjacent; if d0 = . . . = dm−1 = 2 and dm = 3 for 3 ≤ m ≤ k − 1, ∗ then let Uπ,B be obtained from by identifying one end vertex of a path Pm with one ∗ vertex of triangle and the other end vertex of Pm with the root of Tτ,B having SLO*ordering with τ = (dm+1 − 1, dm+2 , · · · , dn−1); if d0 = . . . = dm−1 = 2 and dm ≥ 4 for ∗ 3 ≤ m ≤ k − 1, then let Uπ,B be obtained from by identifying one vertex of a cycle ∗ Cm and the root of Tτ,B having SLO*-ordering with τ = (dm − 2, dm+2 , · · · , dn−1 ). ∗ Clearly, Uπ,B is uniquely determined by π. Then we have the following results Theorem 5.4 [56] For a given graphic unicyclic degree sequence π = (d0 , d1 , . . . , dn−1), with 3 ≤ d0 ≤ . . . ≤ dk and dk+1 = · · · = dn−1 = 1, let G = (V0 ∪ ∂V, E0 ∪ ∂E) be a graph with the Faber-Krahn property in Uπ . Then G has an SLO*-ordering consistent with the first eigenfunction f of G in such a way that v ≺ u implies f (v) ≥ f (u). Theorem 5.5 [56] For a given graphic unicyclic degree sequence π = (d0 , d1 , . . . , dn−1) ∗ with 3 ≤ d0 ≤ . . . ≤ dk and dk+1 = . . . = dn−1 = 1, then λD (G) ≥ λD (Uπ,B ) with ∗ ∗ equality if and only if G = Uπ,B . In other words, Uπ,B is the unique uncyclic graph with the Faber-Krahn property in Uπ , which can be regarded as ball approximation. For d0 = 2, we proposed the following conjecture. Conjecture 5.6 [56] Let π = (d0 , d1 , . . . , dk−1, 1, . . . , 1) be a graphic unicyclic degree sequence with 2 ≤ d0 ≤ d1 ≤ . . . ≤ dk−1 and dk = . . . = dn−1 = 1. Then λD (G) ≥ ∗ ∗ ∗ λD (Uπ,B ) with equality if and only if G = Uπ,B . In other words, Uπ,B is the unique graph with the Faber-Krahn property in Uπ . 15

∗ Theorem 5.7 [58] For a given positive integer k ≤ n − 3, let Un,k be unicyclic graph obtained from by identifying one end vertex of a path Pn−k−1 and one vertex of a triangle and the other end vertex of Pn−k−1 and the center of star K1,k . Then for any ∗ unicyclic graph G of order n with k pendant vertices, λD (G) ≥ λD (Un,k ) with equality ∗ if and only if G = Un,k . In other words, U ∗ is the only unicyclic graph with the Faber-Krahn property in the set Uk of all unicyclic graphs of order n with k pendant vertices, which can be regarded a ball.

6

Graphic degree sequences

In this section, we discuss some extremal properties of Gπ . Theorem 6.1 ([4],[61])For a given graphic degree sequence π, if G is a simple connected graph that has the largest (resp. signless Laplacian) spectral radius in Gπ , then G has a BFS-ordering, i.e., G is a BFS graph. Moreover, all extremal graphs having the (signless Laplacian) spectral radius for bicyclic (tricyclic) degree sequence have been characterized (see [26],[28], [33]). On the Dirichlet eigenvalues for bicyclic degree sequences, we have the following results. For given a graphic bicyclic degree sequence π = (d0 , · · · , dn−1 ) with 3 ≤ d0 ≤ · · · dk−1 and dk = · · · = dn−1 = 1, let G∗π,B be the unique graph having SLO* ordering v0 v1 · · · vn−1 with the induced subgraph by {v0 , v1 , v2 , v3 } being K4 − v2 v3 . Then Theorem 6.2 [57] Let π = (d0 , d1 , · · · , dk−1, 1, 1, · · · , 1) be a graphic bicyclic degree sequence with 3 ≤ d0 ≤ d1 · · · ≤ dk−1 for k ≥ 4. Then for any bicyclic graph G with degree sequence π, λD (G) ≥ λD (Gπ,B ) with equality if and only if G = G∗π,B . In other words, G∗π,B is the only extremal graph with the smallest first Dirichlet eigenvalue in Gπ . Theorem 6.3 [57] Let π = (d0 , d1 , · · · , dk−1, 1, 1, · · · , 1) and τ = (d′0 , d′1 , · · · , d′k−1, 1, 1, · · · , 1) be two different graphic bicyclic degree sequences. If π ⊳ τ with d0 ≥ 3, d′0 ≥ 3, then λD (Gπ,B ) > λD (Gτ,B ). A graph H is a minor of a graph G if H can be obtained from a subgraph of G by contracting edges. An H minor is a minor isomorphic to H. If H is a complete graph, we say that G contains a clique minor of size |H|. For a graph G, the Hadwiger 16

number h(G) of G is the maximum integer k such that G contains a clique minor of size k. Moreover, denote by χ(G) the chromatic number of G. Let h(π) = max{h(G) : G ∈ Gπ }, χ(π) = max{χ(G) : G ∈ Gπ }.

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Conjecture 6.4 (Hadwigers Conjecture for Degree Sequences, [39]) For every graphic degree sequence π, χ(π) ≤ h(π). Recently, Dvo˘rák and Mohar proved a stronger result than the above conjecture. Theorem 6.5 [16] For every graphic degree sequence π, χ(π) ≤ h′ (π) ≤ h(π),

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where h′ (G) is the maximum k such that G has a Kk - topological minor and χ′(π) = max{h′ (G) : G ∈ Gπ }.

References [1] E. O. D. Andriantiana, Energy, Hosoya index and Merrifield-Simmons index of trees with prescribed degree sequence, Discrete Appl. Math., 161 (2013) 724-741. [2] F. Belardo, E. M. Marzi, S. K. Simć and J. F. Wang, On the spectral radius of unicyclic graphs with presribed degree sequence, Linear Algebra Appl., 432(2010) 2323-2334. [3] T. Bıyıko˘glu and J. Leydold, Faber-Krahn type inequalities for trees, J. Combin. Theory Ser. B, 97(2007) 159-174. [4] T. Bıyıko˘glu and J. Leydold, Graphs with given degree sequence and maximal spectral radius. Electron. J. Combin., 15(2008) Paper 119. [5] T. Bıyıko˘glu, M. Hellmuth, and J. Leydold. Largest eigenvalues of the discrete p-Laplacian of trees with degree sequence. Electron. J. Linear Algebra, 18(2009) 202-210. [6] T. Bıyıko˘glu and J. Leydold, Semiregular trees with minimal Laplacian spectral radius, Linear Algebra Appl., 432(2010) 2335-2341. 17

[7] B. bollobás, Extremal Graph Theory, Academic Press, New york, 1978. [8] B. Bollobás, Extremal graph theory. Handbook of combinatorics, Vol. 1, 2, 12311292, Elsevier, Amsterdam, 1995 [9] J. A. Bondy and U. S. R. Murty, Graph theory with applications, Macmillan Press, New York, 1976. [10] F. R. K. Chung, Spectral Graph Theory, AMS Publications, 1997. [11] F. Chung and L.Lu, The average distance in random graphs with given expected degree, Proceeding of National Academy of Science, 99 (2002) 15879-15882. [12] F. Chung and L. Lu, Complex graphs and networks, AMS Publications, Providence, 2006. [13] D. Cvetković, M. Doob and H. Sachs, Spectra of Graphs- Theory and Applications, Academic Press, New Work,1980. Third edition, 1995. [14] D. Cvetković, P. Rowlinson and S. Simić, Signless Laplacian of finite graphs, Linear Algebra Appl., 423(2007) 155-171. [15] A. A. Dobrynin, R. Entringer and I. Gutman, Wiener index of trees: theory and applications, Acta Appl. Math. 66(2001) 211-249. [16] Z. Dvo˘rák and B. Mohar, Chromatic number and complete graph substructures for degree sequences, Combinatorica, to appear. [17] P. Erdös and T. Gallai, Graphs with prescribed degrees of vertices (Hungarian) Mat. Lapok, 11(1960) 264-274. [18] P. Erdös, M. S. Jacobson and J. Lehel, Graphs realizing the same degree sequences and their respective clique numbers, Graph theory, combinatorics, and applications, Vol. 1 (Kalamazoo, MI, 1988), 439 -449, Wiley-Intersci. Publ., Wiley, New York, 1991. [19] M. Ferrara, Some Problems on Graphic Sequences, submitted. [20] M. Fischermann, A. Hoffmann, D. Rautenbach, L. Szekely and L. Volkmann, Wiener index versus maximum degree in trees, Discrete Appl. Math., 122(2002) 127-137. 18

[21] J. Friedman, Some geometric aspects of graphs and their eigenfunctions, Duke Math. J., 69(1993) 487-525. [22] J. M. Guo, On the Laplacian spectral radius of a tree, Linear Algebra Appl., 368(2003) 379-385. [23] I. Gutman and N. Trinajstić, Graph theory and molecular orbitals. Total πelectron energy of alternant hydrocarbons, Chem. Phys. Lett. 17 (1972) 535-538. [24] S. L. Hakimi, On realizability of a set of integers as degrees of the vertices of a linear graph, SIAM Appl. Math., 10 (1962) 496-506. [25] H. Hosoya, Topological index, A newly proposed quantity characterizing the topological nature of structural isomers of saturated hydrocarbons, Bull. Chem. Soc. Jpn. 4(1971) 2332-2339. [26] Y. F. Huang, B. L. Liu and Y. L. Liu, The signless Laplacian spectral radius of bicyclic graphs with prescribed degree sequences. Discrete Math., 311 (2011) 504 -511. [27] F. Jelen and E. Triesch, Superdominance order and distance of trees with bounded maximum degree, Discrete Appl. Math., 125(2003) 225-233. [28] X. Jiang, Y. Liu, B. L. Liu, A further result on majorization theorem, Linear Multilinear Algebra, 59 (2011) 957-967. [29] D. J. Klein, I. Lukovits and I. Gutman, On the definition of the hyperCWiener index for cycleCcontaining structures, J. Chem. Inf. Comput. Sci. 35 (1995) 50-52. [30] J. Leydold, A Faber-Krahn-type inequality for regular trees, Geom. Funct. Anal., 7(1997) 364-378. [31] J. Leydold, The geometry of regular trees with the Faber-Krahn property, Discrete Math., 245(2002) 155-172. [32] J. S. Li and J. H. Yin, Extremal graph theory and degree sequences (Chinese), Adv. Math. (China), 33 (2004) 273-283. [33] M. H. Liu, The (signless Laplacian) spectral radii of connected graphs with prescribed degree sequences, Electron. J. Combin., 19(2012), no. 4, Paper 35, 19 pp. 19

[34] L. Lovász, Combinatorial Problems and Exercises, 2nd Edition, North-Holland, Amsterdam, 1993. [35] A. W. Marshall and I. Olkin, Inequalities: Theory of Majorization and Its Applications, Mathematics in Science and engineering, vol.143, Academic Proess, New York, 1979. [36] R. Merris, Laplacian matrices of graphs: a survey, Linear Algebra Appl., 197-198 (1994) 143-176. [37] J. J.Molitierno, Applications of combinatorial matrix theory to Laplacian matrices of graphs, Discrete Mathematics and its Applications (Boca Raton). CRC Press, Boca Raton, FL, 2012. [38] A. R. Pruss, Discrete convolution-rearrangement inequalities and the FaberKrahn inequality on regular trees, Duke Math. J. 91 (1998), no. 3, pp. 463-514. [39] N. Robertson and Z. X. Song, Hadwiger number and chromatic number for near regular degree sequences,J. Graph Theory, 64(2010) 175-183. [40] N. Schmuck, S. Wagner, H. Wang, Greedy trees, caterpillars, and Wiener-type graph invariants, MATCH Commun. Math. Comput. Chem., 68 (2012) 273-292. [41] J. K. Senior, Partitions and their representative graphs, Amer. J. Math., 73 (1951)663-689. [42] G. Sierksma and H. Hoogeveen, Seven criteria for integer sequences being graphic, J. Graph Theory, 15 (1991) 223-231. [43] R. Shi, The average distance of trees, Systems Science and Mathematical Sciences, 6(1)(1993), 18-24. [44] D. Stevanović, Bounding the largest eigenvalue of trees interms of the largest vertex degree, Linear Algebra Appl. 360(2003) 35-42. [45] L. A. Székely and H. Wang, On subtrees of trees, Adv. in Appl. Math., 34(2005) 138-155. [46] L. A. Székely, H. Wang, and T. Y. Wu, The sum of the distances between the leaves of a tree and the ”semi-regular” property, Discrete Math., 311(2011) 11971203. 20

[47] H. Takeuchi. The spectrum of the p-Laplacian and p-Harmonic morphisms on graphs. Illinois Journal of Mathematics, 47(2003) 939-955. [48] S. W. Tan. On the weighted trees with given degree sequence and positive weight set. Linear Algebra Appl., 433(2010) 380-389. [49] A. Vince and H. Wang, The average order of a subtree of a tree, J. Combin. Theory Ser. B. , 100(2010) 161-170. [50] S. G. Wagner, Correlation of graph-theoretical indeces, SIAM J. Discrete Math., 21(2007) 33-46. [51] S. G. Wagner, H. Wang and X.-D. Zhang, Distance-based graph invariants of trees and the Harary index, Filomat 27 (2013) 41-50. [52] H. Wang, The extremal values of the Wiener index of a tree with given degree sequence, Discrete Appl. Math., 156(2009) 2647-2654. [53] W. D. Wei, The class ℜ(R, S) of (0,1)-matrices, Discrete Math., 39 (1982) 201205. [54] H. Wiener, Structural determination of paraffin boiling points, J. Amer. Chem. Soc., 69(1947), 17-20. [55] G. J. Zhang and X. D. Zhang, The p-Laplacian spectral spectral radius of weighted trees with a degree sequence and a weight set, Electron. J. Linear Algebra, 22(2011) 267-276. [56] G. J. Zhang, J. Zhang and X. D. Zhang, Faber-Krahn Type Inequality for Unicyclic Graphs. Linear Multilinear Algebra, 60(2012) 1355-1364. [57] G. J. Zhang and X. D. Zhang, The first Dirichlet eigenvalue of bicyclic graphs, Czech. Math. J. , 62(2012) 441-451. [58] G. J. Zhang and X. D. Zhang, Faber-Krahn type inequalities for unicyclic graphs with the same boundaries, Chinese Journal of Contemporary Mathematics, to appear. [59] X. D. Zhang, On the two conjectures of Graffiti, Linear Algebra Appl., 385 (2004) 369-379. 21

[60] X.-D. Zhang, The Laplacian spectral radii of trees with degree sequences, Discrete Math., 308(2008) 3143-3150. [61] X.-D. Zhang, The signless Laplacian spectral radius of graphs with given degree sequences, Discrete Appl. Math. 157 (2009), no. 13, pp. 2928-2937. [62] X. D. Zhang and J. S. Li, The Laplacian spectrum of mixed graphs, Linear Algebra Appl., 351(2002) 11-20. [63] X.-D. Zhang, Y. Liu and M.-X. Han, Maximum Wiener index of trees with given degree sequence, MATCH Commun. Math. Comput. Chem., 64 (2010) 661-682. [64] X. D. Zhang, Q. Y. Xiang, L. Q. Xu and R. Y. Pan, The Wiener index of trees with given degree sequences, MATCH Commun. Math. Comput. Chem., 60 (2008) 623-644. [65] X.-M. Zhang, X.-D. Zhang, D. Gray and H. Wang, The number of subtrees with given degree sequence, Journal of Graph Theory, 73(2013) 280-295.

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