Forbidden subgraphs for graphs of bounded spectral radius,
arXiv:1708.02317v1 [math.CO] 7 Aug 2017
with applications to equiangular lines Zilin Jiang∗,†
Alexandr Polyanskii∗,‡
Abstract The spectral radius of a graph is the largest eigenvalue of its adjacency matrix. Let F (λ) be the family of connected graphs of spectral radius ≤p λ. We show that F (λ) can be defined by a finite √ ∗ set of forbidden subgraphs for every λ < λ := 2 + 5 ≈ 2.058, whereas F (λ) cannot for every λ ≥ λ∗ . The study of forbidden subgraphs characterization for F (λ) is motivated by the problem of estimating the maximum cardinality of equiangular lines in the n-dimensional Euclidean space Rn — a family of lines through the origin such that the angle between any pair of them is the same. Denote by Nα (n) the maximum number of equiangular lines in Rn with angle arccos α. We establish an approach to determine the constant c such that Nα (n) = cn + Oα (1) for every 1 1 α > 1+2λ ∗ . We also show that Nα (n) ≤ 1.81n + Oα (1) for every α 6= 3 , which improves a recent result of Balla, Dr¨ axler, Keevash and Sudakov.
1
Introduction
The spectral radius of a graph G, denoted by λ1 (G), is the largest eigenvalue of its adjacency matrix. Let F(λ) be the family of connected graphs of spectral radius ≤ λ. It is well known that λ1 is monotone in the sense that λ1 (G1 ) ≤ λ1 (G2 ) if G1 is a subgraph of G2 , moreover λ1 (G1 ) < λ1 (G2 ) if G1 is a proper subgraph of a connected graph G2 . This implies that F(λ) is closed under taking subgraphs. It is natural to ask if F(λ) can be defined by a finite set of forbidden subgraphs. We determine the threshold λ∗ below which the answer is yes. p √ Theorem 1. For every λ < λ∗ := 2 + 5, there exist finitely many graphs G1 , G2 , . . . , Gn such that F(λ) consists exactly of the connected graphs which do not contain any of G1 , G2 , . . . , Gn as a subgraph. However, the same conclusion does not hold for every λ ≥ λ∗ . The motivation to understand the forbidden subgraphs characterization for F(λ) comes from the problem of estimating the maximum cardinality N (n) of equiangular lines in the n-dimensional ∗
Department of Mathematics, the Technion – Israel Institute of Technology, Technion City, Haifa 32000. Email:
[email protected]. Supported in part by ISF grant nos 1162/15, 936/16. ‡ Moscow Institute of Physics and Technology and Institute for Information Transmission Problems RAS. Email:
[email protected]. Supported in part by ISF grant no. 409/16, and by the Russian Foundation for Basic Research through grant nos 15-01-99563 A, 15-01-03530 A. †
1
Euclidean space Rn — a family of lines through the origin such that the angle between any pair of them is the same. It is considered to be one of the founding problems of algebraic graph theory to determine N (n). The “absolute bound” N (n) ≤ n+1 was established by Gerzon (see [LS73, Theorem 2 3.5]). A remarkable construction of de Caen [dC00] shows that N (n) ≥ 29 (n + 1)2 for n of the form n = 6 · 4k − 1 (see [GKMS16] for a generalization and [JW15] for an alternative constructions). In these constructions, the common angle tends to π/2 as dimension grows. The question of determining the maximum number Nα (n) of equiangular lines in Rn with common angle arccos α was first raised by Lemmens and Seidel [LS73] in 1973, who showed that N1/3 (n) = 2n − 2 for n ≥ 15 and also conjectured that N1/5 (n) equals ⌊3(n − 1)/2⌋ for sufficiently large n. This was later confirmed by Neumaier [Neu89] (see also [GKMS16]). Besides, the “relative bound” 1−α2 2 (see [vLS66, Lemma 6.1] and [LS73, Nα (n) ≤ 1−nα 2 · n is only valid in small dimensions n < 1/α Theorem 3.6]). A general bound due to Neumann [LS73, Theorem 3.4] states that Nα (n) ≤ 2n unless 1/α is an odd integer. For many years, a linear upper bound for Nα (n) was not known when 1/α is an odd integer bigger than 5. An important progress was recently made by Bukh [Buk16], who proved that 2 Nα (n) ≤ cα n for some cα = 2O(1/α ) . Subsequently, Balla, Dr¨axler, Keevash and Sudakov [BDKS17] drastically improved the upper bound to Nα (n) ≤ 1.93n for sufficiently large n relative to α whenever α 6= 13 . We combine Theorem 1 and the framework developed in [BDKS17] to establish a result of the 1 form Nα (n) = cα n + O(1) for all α > α∗ := 1+2λ ∗. Theorem 2. Suppose α > α∗ :=
√1
1+2
√ . 2+ 5
The maximum number Nα (n) of equiangular lines in
α · n + O(1), where kα is the smallest k such that Rn with angle arccos α equals kαk−1 radius of a graph on k vertices. If kα = ∞, then Nα (n) = n + O(1).
1−α 2α
is the spectral
Remark 1. Throughout the paper, as the big-O and little-o notations all depend on α, we suppress the subscripts in Oα (·) and oα (·). Applying the above theorem to the connected graphs with ≤ 3 vertices,
√ whose spectral radii are 1, 2 and 2 respectively, we obtain the old results N1/3 (n) = 2n + O(1), N1/5 (n) = 32 n + O(1) and surprisingly a new result N1/(1+2√2) (n) = 23 n + O(1). This refutes the second part of Conjecture 6.1 in [BDKS17]. The first part of the conjecture, which was also raised by Bukh [Buk16, Conjecture 8], says the following. Conjecture 1. The maximum number of equiangular lines in Rn with angle arccos α equals O(1) if 1/α is an odd positive integer. Our results motivate the following stronger conjecture. 2
1+α 1−α
·n+
Conjecture 2. The maximum number Nα (n) of equiangular lines in Rn with angle arccos α equals kα kα −1 · n + O(1), where kα is defined as in Theorem 2. If kα = ∞, then Nα (n) = n + O(1). The rest of the paper is organized as follows. In Section 2, we prove Theorem 1. In Section 3.1, we review the framework developed by Balla et al.. In Section 3.2, we apply Theorem 1 to estimate Nα (n) for α > α∗ , and in Section 3.3, we extrapolate our method to obtain an upper bound on Nα (n) for α ≤ α∗ , from which follows an improved upper bound Nα (n) ≤ 1.81n + O(1). In the concluding section we discuss evidences supporting Conjecture 2 and a possible extension of our method.
Forbidden subgraphs of F (λ)
2
Suppose there is a finite forbidden subgraphs characterization, say G1 , G2 , . . . , Gn for F(λ). By the monotonicity of spectral radius, we know that no graph has spectral radius in (λ, λ′ ), where λ′ = mini (λ1 (Gi )). Let Λ1 consist of the spectral radius of all graphs or all orders, and denote by lim Λ1 the set of limit points of Λ1 and lim+ Λ1 := {λ ∈ R : (λ, λ + ε) ∩ Λ1 6= ∅ for all ε > 0} the set of right-sided limit points of Λ1 . The contrapositive of the above observation says that F(λ) does not have a finite forbidden subgraphs characterization for all λ ∈ lim+ Λ1 . Hoffman was interested in a related set R consisting of the spectral radius of all symmetric matrices of all orders with non-negative integer entries, and he proved the following theorem on lim R. Theorem 3 (Hoffman [Hof72]). For n = 1, 2, . . . , let βn be the positive root of xn+1 = 1 + x + · · · + xn−1 . 1/2
Let αn = βn
−1/2
+ βn
. Then 2 = α1 < α2 < . . . are all limit points of R smaller than limn αn = λ∗ .
In fact, Hoffman proved the above theorem by first showing [Hof72, Proposition 2.1] that Λ1 = R. He also computed the limit of spectral radii of several families of graphs. We compile some of his computation and other relavant results in the following lemma, the proof of which is presented in Appendix A. We use the notation αn ր α if α1 < α2 < . . . and limn αn = α, and αn ց α if α1 > α2 > . . . and limn αn = α. Lemma 4. Let αn be defined as in Theorem 3 and let β ∗ ≈ 2.2225 be the positive root of x6 − 7x4 + 12x2 − 9. Denote by Cn the cycle with n vertices, Pn the path with n vertices and Sn the star with √ n leaves. Define An , Bm,n , Dn , Em,n , Fn , Tn as below. (a)1 λ1 (An ) ր 3/ 2, (b) λ1 (Bm,n ) ց αn for fixed n, (c) λ1 (Cn ) = 2, (d) λ1 (Dn ) ց λ∗ , (e) λ1 (Em,n ) ր αm for fixed m, (f ) λ1 (Fn ) ր λ∗ , (p) √ λ1 (Pn ) ր 2, (s) λ1 (Sn ) = n, (t) λ1 (Tn ) ր β ∗ . The work of finding all the limit points of Λ1 was completed by Shearer. √ It was asserted that λ1 (An ) ր 4/ 3 in [Hof72]. This is a mistake but it will not affect the main result of [Hof72] as long as the limit is > λ∗ . 1
3
An =
Dn =
Bm,n = n
m
n
Fn =
Em,n = m
n
Tn = n
n
n
Theorem 5 (Shearer [She89]). For any λ ≥ λ∗ , there exists a sequence of distinct graphs G1 , G2 , . . . such that lim λ1 (Gi ) = λ. Note that Theorem 5 implies that lim+ Λ1 ⊃ [λ∗ , ∞). Thus by the observation at the beginning of the section, the second half of Theorem 1 is proved. Proof of the first half of Theorem 1. We break the proof into two cases: Case 1: λ < 2. Note that S4 ∈ / F(λ) and Pn ∈ / F(λ) for some n. Clearly, a connected graph G that contains neither S4 nor Pn has at most 4n vertices. Therefore {S4 , Pn } ∪ {connected graph G ∈ / F(λ) with ≤ 4n vertices} is a finite forbidden subgraphs characterization for F(λ). Case 2: 2 ≤ λ < λ∗ . Choose m ≥ 2 such that αm−1 ≤ λ < αm , where αm is as defined in Theorem 3. Then choose n such that An , Em,n , Fn ∈ / F(λ). Note that S5 ∈ / F(λ), B1,m , B2,m , . . . , Bm,m ∈ / F(λ) and D2 , D3 , . . . , Dm+n ∈ / F(λ). We claim that if a connected graph G contains none of G0 := {S5 , An , B1,n , B2,n , . . . , Bm,n , D2 , D3 , . . . , Dm+n , Em,n , Fn } , then G is a path, a cycle, Ei,j for some i < m, or its number of vertices is bounded by a constant, say b, which will be determined by the argument below. In fact, because G does not contain D2 , . . . , Dm+n , Em,n , G does not contain D2 , D3 , . . . , and so G must be a tree or a cycle. We may assume that G is a tree but not a path. As G does not contain S5 , G is a tree of maximum degree ≤ 4. If the maximum degree is indeed 4, then G would contain An when G has sufficiently many vertices. Hereafter, we may assume that the maximum degree of G is 3 exactly. Because G does not contain Fn , when G has sufficiently many vertices, every vertex of degree 3 will be adjacent to a leave, in other words, G is a caterpillar tree of maximum degree 3. Moreover, since G does not contain B1,m , B2,m , . . . , Bm,n , Em,n , G has only one vertex of degree 3, hence G must be one of Ei,j for i < m. Notice that the spectral radii of paths and cycles are at most 2 ≤ λ and the spectral radius λ1 (Ei,j ) < αi ≤ αm−1 ≤ λ for all i < m. Therefore the claim implies that G0 ∪ {connected graph G ∈ / F(λ) with ≤ b vertices} is a finite forbidden subgraphs characterization for F(λ).
4
3 3.1
Equiangular lines The framework to estimate Nα (n)
We shall set the ground by briefly reviewing the framework to estimate Nα (n) developed in [BDKS17]. We advise the readers who are interested in the details of the framework to read at least Section 2.1 of [BDKS17]. Definition 1. Let L be a subset of the interval [−1, 1). A finite non-empty set C of unit vectors in Rn is called a spherical L-code if hv1 , v2 i ∈ L for any pair of distinct vectors v1 , v2 in C. The Gram matrix MC are given by (MC )i,j = hvi , vj i, and the underlying graph GC is defined as follows: let C be its vertex set and for any distinct vi , vj ∈ C, we put the edge (vi , vj ) if and only if hvi , vj i < 0. By choosing a unit vector in the direction of each line in a set of equiangular lines with angle arccos α in Rn , we obtain a spherical {±α}-code C = {v1 , . . . , vm } in Rn . One can show that the clique number of GC is ≤ 1 + α−1 . By Ramsey’s theorem, if |C| is large enough, then GC contains an independent set of size, say t. Assume that I = {v1 , . . . , vt } ⊂ C is an independent set in GC . By properly replacing vi by −vi for some i > t, we may assume that the degree of vi to I is at most t/2 for all i > t. One can then show that the number of vertices that are not independent from I in GC is bounded by b = Oα,t (1). Without loss of generality, assume that each vertex in {vb+1 , . . . , vm } is independent from I. Denote by vi′ the normalized projection of vi onto the orthogonal complement ′ ′ , . . . , vm of span(I). It can be shown that C ′ := vb+1 is a spherical L(α, t)-code, where n L(α, t) := −σ · 1 −
1 t+α−1
, t+α1 −1
o
and σ :=
2α . 1−α
We wrap up the above discussion in the following lemma, which is essentially [BDKS17, Lemma 2.8]. The slight difference in the statement originates from our need to estimate Nα (n) up to a constant error relative to α. However, the same proof goes through without alternation. Lemma 6 (Lemma 2.8 of Balla et al. [BDKS17]). Let α ∈ (0, 1) and t ∈ N be fixed. For any spherical {±α}-code C in Rn , there exists a spherical L(α, t)-code C ′ in Rn such that |C| ≤ |C ′ | + Oα,t (1). As t goes to ∞, L(α, t) approaches {−σ, 0}. One of the connections between spherical {±α}-codes and spherical {−σ, 0}-codes is the following. We shall denote by In the identity matrix of order n and Jn the all-ones matrix of order n, and we surpress the subscripts when the order of the matrices are clear from the context. Lemma 7. Let α ∈ (0, 1) and σ = 2α/(1−α). For any spherical {−σ, 0}-code C0 in Rk , there exists a spherical {±α}-code C in Rn such that |C| ≥ ⌊ n−1 k ⌋ |C0 |. In particular, the maximum number Nα (n) n of equiangular lines in R with angle arccos α is ≥ ⌊ kn−1 ⌋kα , where kα is defined as in Theorem 2. α −1 In the case kα = ∞, we have Nα (n) ≥ n. 5
Proof. Given a spherical {−σ, 0}-code C0 in Rk . Let m = ⌊ n−1 k ⌋ and M0 be the Gram matrix of C0 . Consider the matrix M := (1− α)M0 ⊗ Im + αJ of order m |C0 |. One can check that M is semipositive definite and rank(M ) ≤ m · rank(C0 ) + 1 ≤ mk + 1 ≤ n. Moreover, the diagonal entries are ones and the off-diagonal entries are either −α or α. Therefore M can be realized as the Gram matrix of a spherical {±α}-code in Rn of size m |C0 |. When kα < ∞, it suffices to construct a spherical {−σ, 0}-code C0 in Rkα −1 of size kα . Because 1/σ = 1−α 2α is the spectral radius of a graph G on kα vertices, I − σA is positive semidefinite, where A is the adjacency matrix of G. Moreover, rank(I − σA) ≤ kα − 1. Clearly, I − σA can be realized as the Gram matrix of a spherical {−σ, 0}-code in Rkα −1 of size kα . When kα = ∞, one can check that (1 − α)In + αJn can be realized as the Gram matrix of a spherical {±α}-code in Rn of size n.
3.2
Application of Theorem 1
We recall two classical results — a fact about the spectral radius of a connected graph and a necessary condition on eigenvalues of the sum of two matrices. Theorem 8 (Corollary of Perron–Frobenius theorem [Fro12, Per07]). For every connected graph G, λ1 (G) has multiplicity 1. Theorem 9 (Weyl’s inequality [Wey12]). Given two n × n Hermitian matrices A and B. Denote the eigenvalues of A as λ1 (A) ≥ λ2 (A) ≥ · · · ≥ λn (A), and similarly denote the eigenvalues of B and A + B. Whenever 0 ≤ i, j, i + j < n, λi+j+1 (A + B) ≤ λi+1 (A) + λj+1 (B). The motivation of the proof for Theorem 2 comes from the following observation. Suppose C ′ is a spherical L(α, t)-code in Rn . Let MC ′ be its Gram matrix, G′ its underlying graph with adjacency matrix A′ . The matrices MC ′ and A′ are related by 1 1 1+ J. MC ′ = I − σA′ + −1 −1 t+α +1 t+α −1 Since MC ′ is positive semidefinite and one has the freedom to choose t large, it seems plausible that I − σA′ is positive semidefinite as well. In this case, one can easily finish the proof. Unfortunately, the positive eigenvalue of J is |C ′ | that is not bounded. This allows I − σA′ to have a negative eigenvalue, in other words, the spectral radius of G′ is > λ = 1/σ. Theorem 1 roughly says that the reason for λ1 (G′ ) > λ is local, that is, G′ contains a forbidden subgraph of bounded size. Therefore, a priori we can choose t large to get a contradiction from I − σA′ not being positive semidefinite. α Proof of Theorem 2. In view of Lemma 7, it suffices to show that Nα (n) ≤ kαk−1 · n + O(1) and Nα (n) ≤ n + O(1) in the case kα = ∞. Suppose C is a spherical {±α}-code in Rn . We first find out t ∈ N with which we will apply 2α 1 1 1−α 1−α∗ Lemma 6. Denote σ := 1−α and λ := σ1 . Since α > α∗ = 1+2λ ∗ , we have that λ = σ = 2α < 2α∗ = λ∗ . By Theorem 1, there exists a finite family of graphs G such that F(λ) consists exactly of the
6
connected graphs which do not contain any graph in G as a subgraph. Because λ1 (G) > λ = 1/σ for every G ∈ G, we thus choose t large enough so that 1 − σλ1 (G) +
v(G) < 0 for all G ∈ G, t + α−1 − 1
(1)
where v(G) denotes the number of vertices in G. By Lemma 6, there exists a spherical L(α, t)-code C ′ in Rn such that |C| ≤ |C ′ | + O(1). Let MC ′ be the Gram matrix of C ′ , and GC ′ the underlying graph. We decompose GC ′ into m connected components, say G1 , . . . , Gm , with vertex sets C1 , . . . , Cm respectively. We claim that Gi does not contain any graph in G for all i ∈ [m]. Suppose on the contrary that Gi contains a graph G ∈ G. Without loss, assume that i = 1. Let G′1 with vertex set C1′ be the minimal induced subgraph of G1 that contains G as a subgraph. Apparently v(G′1 ) = v(G). By the monotonicity of λ1 and the choice of t in (1), we obtain v(G′1 ) 1 − σλ1 (G′1 ) + < 0. (2) t + α−1 − 1 Let MC1′ be the Gram matrix of C1′ and let A′1 be the adjacency matrix of G′1 . Theses two matrices are related by the equation 1 1 J. MC1′ = I − σA′1 + 1+ −1 −1 t+α −1 t+α −1
Using the fact that λ1 (J) = v(G′1 ) and (2), we know from Weyl’s inequality that the least eigenvalue of MC1′ is negative. This contradicts with the fact that a Gram matrix is positive semidefinite. Because G is a forbidden subgraphs characterization for F(λ), by the claim Gi ∈ F(λ), that is λ1 (Gi ) ≤ λ, for all i ∈ [m]. In other words, I − σAi is positive semidefinite, where Ai is the adjacency matrix of Gi . By Theorem 8, rank(I − σAi ) ≥ |Ci | − 1 and equality holds if and only if λ1 (Ai ) = 1/σ = λ. Finally, we use the relation I − σA1 I − σA2 1 1 + 1+ J MC ′ = . −1 −1 .. t+α −1 t+α −1 I − σAm
to estimate the rank of MC ′
n ≥ rank(MC ′ ) ≥
m X i=1
!
rank(I − σAi )
− 1.
(3)
If kα is finite, we know that if λ1 (Ai ) = λ = 1−α 2α , then |Ci | ≤ kα and so rank(I − σAi ) ≥ (1 − 1/kα ) |Ci |. Therefore we always have rank(I − σAi ) ≥ (1 − 1/kα ) |Ci |. By (3), we obtain ! X m 1 ′ 1 1 |Ci | − 1 = 1 − C −1≥ 1− n ≥ 1− |C| − O(1), kα kα kα i=1
α which is equivalent to the desired inequality after multiplying kαk−1 . Otherwise since kα = ∞, we get Pm ′ rank(I − σAi ) ≥ |Ci | and n ≥ ( i=1 |Ci |) − 1 = |C | − 1 ≥ |C| − O(1).
7
3.3
An improved upper bound on Nα (n)
In this section, we prove Nα (n) ≤ 1.81n + O(1). We need the following spectral techniques. Lemma 10 (Lemma 2.10 of Bella et al. [BDKS17]). Let C be a spherical L(α, t)-code in Rn and let GC be the underlying graph of C of average degree d. Then |C| ≤ (1 + dσ 2 ) · rank(I − σA), where 2α and A is the adjacency matrix of GC . σ = 1−α Lemma 11 (Lemma 2.13 of Balla et al. [BDKS17]). Let G be a graph with minimum degree δ > 1. Let v0 be a vertex of G and let H be the subgraph consisting of all vertices within distance k of v0 . √ 2k Then λ1 (H) ≥ k+1 δ − 1. Again, Lemma 10 is stated differently from [BDKS17, Lemma 2.13], and the same proof works line by line. Theorem 12. Let Nα (n) be the maximum number of equiangular lines in Rn and let σ := Nα (n) ≤ (1 + 3σ 2 )n + O(1) Nα (n) ≤ 1 + 2lσ 2 n + O(1)
2α 1−α .
Then
if σ > 1/β ∗ , if 0 < σ ≤ 1/2,
where β ∗ ≈ 2.2225 is defined as in Lemma 4 and l is the smallest integer >
1 . 4σ2
Proof. Set λ := 1/σ. The stratagem is to find a finite partial forbidden subgraphs characterization G0 ⊂ F(λ)c in the following sense: if a connected graph G does not contain any of G0 , then either G ∈ F(λ) or G has average degree ≤ d for some constant d. We claim that if such G0 exists, then 1 2 , dσ n + O(1). (4) Nα (n) ≤ 1 + max kα − 1 The proof of the claim follows the outline of the proof for Theorem 2. Suppose C is a spherical {±α}-code in Rn . We choose t large enough so that (1) holds for all G ∈ G0 . By Lemma 6, there exists a spherical L(α, t)-code C ′ in Rn such that |C| ≤ |C ′ | + O(1). Define MC ′ , GC ′ , G1 , . . . , Gm , C1 , . . . , Cm , A1 , . . . , Am as in the proof of Theorem 2. The same claim that Gi does not contain any graph in G0 holds for all i ∈ [n]. By our choice of G0 , we know that either λ1 (Gi ) ≤ λ or the average degree of Gi is ≤ d. In the former case, we can show that |Ci | ≤ (1 + kα1−1 ) · rank(I − σAi ). Whereas, in the latter case, we can apply Lemma 10 and get |Ci | ≤ (1 + dσ 2 ) · rank(I − σAi ). Summing up these estimations for |Ci |’s yields (4). Case σ > 1/β ∗ . Let Sn be the star with n leaves and let Tn be defined as in Lemma 4. Since √ λ1 (Sn ) = n, λ1 (Tn ) ր β ∗ , we know λ1 (S5 ) > λ and we choose n large enough so that λ1 (Tn ) > λ. √ √ √ One can compute that the following graphs have spectral radii 21 (1 + 13), 21 (1 + 17) and 5 respectively, all of which are > β ∗ . Suppose a connected graph G contains no S5 , Tn , U1 , U2 , U3 , and has average degree > 3. By a double counting argument with the Cauchy–Schwarz inequality, one can show that there is a pair of 8
U1 =
U2 =
U3 =
neighboring vertices v1 , v2 such that the sum of their degrees is > 6. Since G has maximum degree 4, we may assume that the degree of v1 is 4 and the degree of v2 is 3 or 4. Since G does not contain U1 or U2 , v1 , v2 do not have common neighbors. Therefore the induced subgraphs on the vertices within distance 1 of v1 , v2 must be one of the following two.
Since G does not contain Tn or U3 , we know that G has at most 5n vertices. Thus we can apply the claim above to d = 3 and G0 := {S5 , Tn , U1 , U2 , U3 } ∪ {connected graph G ∈ / F(λ) with ≤ 5n vertices} . As 3σ 2 > 1/2 and kα ≥ 3 unless α = 1/3, dσ 2 subsumes
1 kα −1
in (4).
Case 0 < σ ≤ 1/2. Choose D ∈ N such that SD ∈ / F(λ). Suppose G has average degree d¯ > 2l and it has maximum degree < D. It is well known that a graph with average degree d¯ contains a ¯ subgraph with minimum degree d/2. Hence G contains a subgraph G′ with minimum degree at least √ 2 2k l > λ. Applying Lemma 11 to l + 1. As l > 4σ1 2 = λ4 ≥ 1, one can choose k large enough so that k+1 √ 2k ′ ′ G , we know that for every vertex v0 of G and k ∈ N, λ1 (H) ≥ k+1 l > λ, where H is the subgraph consisting of all vertices within distance k of v0 . This means that G contains a subgraph H ∈ / F(λ) k on ≤ D vertices. Thus we can apply the claim above to d = 2l and n o / F(λ) with ≤ D k vertices . G0 := {SD } ∪ connected graph G ∈ Again in this case dσ 2 > 1/2 subsumes
1 kα −1
in (4).
Remark 2. In the proof of Theorem 12, Lemma 11 helped provide a local certificate, that is a subgraph p of bounded size, for a graph of high average degree d to have large eigenvalue approximately 2 d/2 − 1. Although it seems possible to refine Lemma 11, such an refinement has a limit in terms of improving the proof of Theorem 12 due to the existence of regular graphs of high girth. The Ramanujan graphs constructed independently by Margulis [Mar82] and Lubotzky, Phillips and Sarnak [LPS88] are dregular graphs on n vertices of girth Ωd (log n). Therefore, locally, a subgraph of these Ramanujan graphs of bounded size looks like a d-regular tree, whose spectral radius is bounded from above by the √ spectral radius 2 d − 1 of the infinite d-regular tree. For d < λ2 /4 + 1, we will not be able to provide a local certificate for some d-regular graphs, hence we can only hope to apply (4) to d ≈ λ2 /4 and get a bound of the form Nα (n) ≤ 1 + 14 n + O(1). 9
Corollary 13. The maximum number of equiangular lines in Rn with angle arccos α is ≤ 1.81n+O(1) if α 6= 1/3. Proof. On the one hand, Theorem 2 implies that Nα (n) ≤ 23 n + O(1) if σ ∈ (1/λ∗ , ∞) \ {1}. On the other hand, Theorem 12 implies that 3 2 √ Nα (n) ≤ 1 + 3σ n + O(1) ≤ 1 + n + O(1) if σ ∈ (1/β ∗ , 1/λ∗ ], 2+ 5 √ if σ ∈ (1/ 8, 1/β ∗ ], Nα (n) ≤ 1 + 4σ 2 n + O(1) ≤ 1 + (2/β ∗ )2 n + O(1) √ √ l 2 n + O(1) if σ ∈ (1/ 4l, 1/ 4l − 4] and l ≥ 3 Nα (n) ≤ (1 + 2lσ )n + O(1) ≤ 1 + 2l − 2 Finally, note that
4
l 2l−2
≤
3 4
if l ≥ 3 and all the above linear coefficients of n are less than 1.81.
Concluding remarks
Besides Theorem 2 and Lemma 7, we discuss two other evidences supporting Conjecture 2. The key parameter kα in the conjecture is the smallest k such that λ := 1−α 2α is the spectral radius of a graph on k vertices. Clearly, if kα < ∞, then 1. λ is an algebraic integer — it is a root of some monic polynomial with coefficients in Z, 2. λ is totally real — its conjugate elements are in R, 3. λ is the largest among its conjugate elements by Perron–Frobenius theorem. On the converse, Bass, Estes and Guralnick [BEG94, Corollary 0.7] proved that any totally real algebraic integer is the eigenvalue of the adjacency matrix of some regular graph. It would be intersting to study a complete set of necessary conditions for the spectral radius of a graph. Notice that kα ≥ deg α, where deg α denotes the algebraic degree of α. Conjecture 2 predicts that deg α Nα (n) ≤ deg α−1 · n + O(1). This is indeed a cheap bound on Nα (n). Proposition 14. If λ = 1−α 2α is a totally real algebraic integer, then Nα (n) ≤ Otherwise n ≤ Nα (n) ≤ n + 1.
deg α deg α−1
· n + O(1).
Proof. Let C be a spherical {±α}-code in Rn . Let MC be its Gram matrix, GC its underlying graph, and A the adjacency matrix of G. We know that MC = (1 − α)(I − σA) + αJ, where σ = 1/λ. If λ is not a totally real algebraic integer, then λ is not an eigenvalue of A, hence rank(I − σA) = |C| and so n ≥ rank(MC ) ≥ |C| − 1. Together with Lemma 7, we have n ≤ Nα (n) ≤ n + 1. If λ is an algebraic number, then rank(I − σA) ≥ (1 − deg1 λ ) |C| and so n ≥ rank(MC ) ≥ (1 − deg1 λ ) |C| − 1. Lemma 7 and Proposition 14 would imply that Conjecture 2 in the equality case kα = deg α. Note that kα = deg α if and only if λ = 1−α 2α is the spectral radius of a graph with irreducible characteristic polynomial. A result of Mowshowitz [Mow71] (see [GM81, Theorem 3.8] for a generalization) states that a graph with irreducible characteristic polynomial has trivial automorphism group. Such graphs
10
are known as asymmetric graphs. Erd˝os and R´enyi [ER63] showed that asymmetric graphs have at least 6 vertices and there are 8 asymmetric graphs on 6 vertices. Interestingly, these 8 graphs all indeed have irreducible characteristic polynomial. Moreover, their spectral radii are larger than λ∗ , for which Theorem 2 fails to address. Hereafter we assume that λ is a totally real algebraic integer. Suppose λ is not the largest among its conjugate elements. Conjecture 2 thus asserts that Nα (n) = n + O(1). This is indeed the case. Proposition 15. Suppose λ = 1−α 2α is a totally real algebraic integer. If λ is not the largest among its conjugate elements, n ≤ Nα (n) ≤ n + 2. Proof. We denote by λ−i (·) and λi (·) respectively the ith smallest eigenvalue and the ith largest eigenvalue of a matrix. Let λ′ > λ be a conjugate element of λ. Let C be a spherical {±}-code in Rn . Let MC be its Gram matrix, GC its underlying graph, and A the adjacency matrix of G. We know that MC = (1−α)(I −σA)+αJ. Assume for the sake of contradiction that rank(I −σA) ≤ |C|−2, that is, λ is an eigenvalue of A with multiplicity ≥ 2, then 1−σλ′ < 0 is an eigenvalue of I −σA with multiplicity ≥ 2, hence λ−2 (I − σA) < 0. By Weyl’s inequality, 0 ≤ λ−1 (MC ) ≤ (1 − α)λ−2 (I − σA) + αλ2 (J) < 0. This is a contradiction. Therefore rank(I − σA) ≥ |C| − 1 and so n ≥ rank(MC ) ≥ |C| − 2. Together with Lemma 7, we have n ≤ Nα (n) ≤ n + 2. Lastly, we remark on the limit of our method. Our proof strategy would yield Conjecture 2 provided that F(λ) has a finite partial forbidden subgraphs characterization G0 ⊂ F(λ)c in the following sense: if a graph G does not contain any graph in G0 , then either G ∈ F(λ) or λ is an eigenvalue of G with multiplicity ≤ v(G)/kα .
Acknowledgements Thanks to Boris Bukh for introducing equiangular lines to the first author, and to Jun Su for some useful correspondence.
References [BDKS17] Igor Balla, Felix Dr¨ axler, Peter Keevash, and Benny Sudakov. Equiangular lines and spherical codes in euclidean space. Inventiones mathematicae, Jul 2017. arXiv:1606.06620[math.CO]. [BEG94]
Hyman Bass, Dennis R. Estes, and Robert M. Guralnick. Eigenvalues of symmetric matrices and graphs. J. Algebra, 168(2):536–567, 1994.
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Boris Bukh. Bounds on equiangular lines and on related spherical codes. SIAM J. Discrete Math., 30(1):549–554, 2016. arXiv:1508.00136[math.CO].
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D. de Caen. Large equiangular sets of lines in Euclidean space. Electron. J. Combin., 7:Research Paper 55, 3, 2000.
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P. Erd˝os and A. R´enyi. Asymmetric graphs. Acta Math. Acad. Sci. Hungar, 14:295–315, 1963.
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¨ Ferdinand Georg Frobenius. Uber matrizen aus nicht negativen elementen. 1912.
[GKMS16] Gary Greaves, Jacobus H. Koolen, Akihiro Munemasa, and Ferenc Sz¨ oll˝osi. Equiangular lines in Euclidean spaces. J. Combin. Theory Ser. A, 138:208–235, 2016. arXiv:1403.2155[math.CO]. [GM81]
C. D. Godsil and B. D. McKay. Spectral conditions for the reconstructibility of a graph. J. Combin. Theory Ser. B, 30(3):285–289, 1981.
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Alan J. Hoffman. On limit points of spectral radii of non-negative symmetric integral matrices. pages 165–172. Lecture Notes in Math., Vol. 303, 1972.
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Jonathan Jedwab and Amy Wiebe. Large sets of complex and real equiangular lines. J. Combin. Theory Ser. A, 134:98–102, 2015. arXiv:1501.05395[math.CO].
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P. W. H. Lemmens and J. J. Seidel. Equiangular lines. J. Algebra, 24:494–512, 1973.
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G. A. Margulis. Explicit constructions of graphs without short cycles and low density codes. Combinatorica, 2(1):71–78, 1982.
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Abbe Mowshowitz. Graphs, groups and matrices. pages 509–522, 1971.
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J. H. van Lint and J. J. Seidel. Equilateral point sets in elliptic geometry. Nederl. Akad. Wetensch. Proc. Ser. A 69=Indag. Math., 28:335–348, 1966.
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Hermann Weyl. Das asymptotische Verteilungsgesetz der Eigenwerte linearer partieller Differentialgleichungen (mit einer Anwendung auf die Theorie der Hohlraumstrahlung). Math. Ann., 71(4):441–479, 1912.
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A
Proof of Lemma 4
Proof of Lemma 4(c). Clearly, for every regular graph, its spectral radius equals its degree. Proof of Lemma 4(p). Note that λ1 (Pn ) is at least the average degree 2 − 2/n of Pn , and λ1 (Pn ) < λ1 (Cn ) = 2. Proof of Lemma 4(s). It follows from the characteristic polynomial of Sn , which is xn−1 (x2 − n). For the proof of other facts in Lemma 4, we shall use the following lemmas due to Hoffman. Lemma 16 (Lemma 3.4 of Hoffman [Hof72]). Let A−1 be a principal submatrix of order n − 1 of a symmetric matrix A0 of order n with non-negative entries. Define Ai+1 recursively by ! Ai eTi , where ei = 0 0 . . . 0 1 . Ai+1 = ei 0 Assume further that limi→∞ λ1 (Ai ) > 2. Then limi→∞ λ1 (Ai ) is the largest positive root of ! √ x + x2 − 4 p0 (x) = p−1 (x), 2
(5)
where pi is the characteristic polynomial of Ai for i = −1, 0. Definition 2. Let G be a connected graph, and let v be a vertex of G. Denote (G, v, n) the graph obtained from G by appending a path of n vertices to G at v. Let G1 , G2 be disjoint connected graphs, and let v1 , v2 be vertices of G1 , G2 respectively. Define (G1 , v1 , n, v2 , G2 ) to be the graph obtained from G1 and G2 by joining them by a path of n vertices connecting v1 and v2 . (G, v, n) =
G
v n
(G1 , v1 , n, v2 , G2 ) =
G1 v1
v2 G2 n
Remark 3. When we apply Lemma 16 to the adjacency matrix of a graph, we get the following interpretation. Let G be a connected graph, and let v be a vertex of G. Assume further that λ1 (G, v, n) ≥ 2 for some n, then lim λ1 (An ) is the largest positive root of (5), where p−1 , p0 are the characteristic polynomials of G \ {v} and G respectively. Lemma 17 (Proposition 4.2 of Hoffman [Hof72]). Let G1 , G2 be disjoint connected graphs, v1 , v2 vertices of degree ≥ 2 of G1 , G2 respectively. Then lim λ1 (G1 , v1 , n, v2 , G2 ) = max {lim λ1 (G1 , v1 , n) , lim λ1 (G2 , v2 , n)} . 13
Definition 3. Let e be an edge of a graph G. If there exists a path in G, x1 , x2 , . . . , xk where xk−1 and xk are the end vertices of e, and the degrees of x1 , x2 , . . . , xk−1 are respectively 1, 2, 2, . . . , 2, then e is said to be on an end path of G. Lemma 18 (Proposition 4.1 of Hoffman [Hof72]). Let G be a connected graph with λ1 (G) > 2, e = (x, y) an edge of G not on an end path of G, G not a cycle. Let G+ e be the graph obtained from G by deleting edge e, and adding a vertex z adjacent to x and y only. Then λ1 (G+ e ) < λ1 (G). We also need the characteristic polynomials for paths and cycles. The readers are invited to prove them by reduction and induction. Lemma 19. Denote pn and qn the characteristic polynomials of Pn and Cn respectively. Then p0 (x) = 1,
p1 (x) = x,
pn = xpn−1 (x) − pn−2 (x),
qn+1 (x) = pn+1 (x) − pn−1 (x) − 2, for all n ≥ 2.
Moreover, the recursion gives pn (x) = where θ = θ(x) :=
√ x+ x2 −4 . 2
θn θ −n + , 1 − θ −2 1 − θ 2
qn (x) = θ n + θ −n ,
(6)
By the monotonicity of λ1 and Lemma 18, the monotonicity of the spectral radii of each family of graphs in Lemma 4 follows immediately. We only need to compute the limit for each family. Proof of Lemma 4(a). Note that An = (S3 , v, n), where v is the vertex of degree 3 in S3 . Note that λ1 (A1 ) = 2. By Remark 3, lim λ1 (An ) is the largest positive root of ! √ x + x2 − 4 x2 (x2 − 3) = x3 , 2 √ which turns out to be 3/ 2. Proof of Lemma 4(f ). Note that Fn = (P5 , v, n), where v is the third vertex of P5 . Observe that λ1 (F2 ) = 2. By Remark 3, lim λ1 (Fn ) is the largest positive root of ! √ 2 x + x2 − 4 x5 − 4x3 + 3x = x2 − 1 , 2 which turns to be the positive root λ∗ of x4 − 4x2 − 1. Proof of Lemma 4(t). Similarly, as λ1 (T1 ) > λ1 (S4 ) = 2, by Remark 3, lim λ1 (Tn ) is the largest positive root of ! √ x + x2 − 4 x6 − 5x4 + 3x2 = x3 x2 − 3 , 2 which turns out to be the positive root β ∗ of x6 − 7x4 + 12x2 − 9. 14
Proof of Lemma 4(e). For the m = 1 case, it is well known that λ1 (E1,n ) is at least the average degree 2 + 4/n and so lim λ1 (E1,n ) ≥ 2. On the other hand, assume for the sake of contradiction that lim λ1 (E1,n ) > 2. By Remark 3, lim λ1 (E1,n ) is the largest positive root of ! √ x + x2 − 4 x x2 − 2 = x2 , 2 which turned out to be 2 contradicting the assumption lim λ1 (E1,n ) > 2. For the m ≥ 2 case, because λ1 (Em,8 ) ≥ λ1 (E2,8 ) = 2 and Em,n = (Pm+2 , v, n), where v is the second vertex of Pm+2 , Remark 3 and Lemma 19 gives that limn→∞ Em,n is the largest positive root of θpm+2 (x) = xpm (x), where θ and pi are defined as in Lemma 19. From this and (6), using x = θ + 1/θ, z = θ 2 , we seek the largest root of z m+1 = 1 + z + · · · + z m−1 . In view of Theorem 3, this proves that z = βm and limn→∞ λ1 (Em,n ) = αm . Proof of Lemma 4(b). Let v1 be the vertex in S2 of degree 2 and v2 be second vertex in Pn+2 . By Lemma 17, n o lim λ1 (Bm,n ) = max lim λ1 (S2 , v1 , m), lim λ1 (Pn+2 , v2 , m) m→∞ m→∞ nm→∞ o = max lim λ1 (E1,m ), lim λ1 (En,m ) = αn . m→∞
m→∞
Proof of Lemma 4(d). Let Mn be the adjacency matrix of Dn and let rn be its characteristic polynomial. By expanding the determinant of xI − Mn along the row of the leaf of Dn , one obtains rn (x) = xqn+1 (x) − pn (x),
where pn and qn+1 are defined as in Lemma 19. Using x = θ + 1/θ and z = θ 2 , we seek the largest √ 1+ 5 2 −(n+1) 2 root of z − z − 1 = z . As z > 0, we get z − z − 1 > 0, hence z > φ, where φ = 2 is the golden ratio. As n → ∞, the largest root tends to φ, and so λ1 (Dn ) ց φ1/2 + φ−1/2 = λ∗ .
15
