Spherical Two-Distance Sets and Eigenvalues of Signed Graphs

We study the problem of determining the maximum size of a spherical two-distance set with two fixed angles (one acute and one obtuse) in high dimensions. Let Nα,β(d)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$N_{\alpha ,\beta }(d)$$\end{document} denote the maximum number of unit vectors in Rd\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathbb {R}}^d$$\end{document} where all pairwise inner products lie in {α,β}\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\{\alpha ,\beta \}$$\end{document}. For fixed -1≤β<0≤α<1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$-1\le \beta<0\le \alpha <1$$\end{document}, we propose a conjecture for the limit of Nα,β(d)/d\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$N_{\alpha ,\beta }(d)/d$$\end{document} as d→∞\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$d \rightarrow \infty $$\end{document} in terms of eigenvalue multiplicities of signed graphs. We determine this limit when α+2β<0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\alpha +2\beta <0$$\end{document} or (1-α)/(α-β)∈{1,2,3}\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(1-\alpha )/(\alpha -\beta ) \in \{1, \sqrt{2}, \sqrt{3}\}$$\end{document}. Our work builds on our recent resolution of the problem in the case of α=-β\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\alpha = -\beta $$\end{document} (corresponding to equiangular lines). It is the first determination of limd→∞Nα,β(d)/d\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\lim _{d \rightarrow \infty } N_{\alpha ,\beta }(d)/d$$\end{document} for any nontrivial fixed values of α\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\alpha $$\end{document} and β\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\beta $$\end{document} outside of the equiangular lines setting.


Introduction
A set of unit vectors in R d is a spherical two-distance set if the inner products of distinct vectors only take two values.The problem of determining the maximum size of spherical two-distance sets is a deep and natural problem in discrete geometry.Some of the earliest results in this area date to the seminal work of Delsarte, Goethals, and Seidel [4].They prove that a spherical two-distance set in R d has size at most 1  2 d(d + 3).This bound is close to the truth, as taking the 1 2 d(d + 1) midpoints on the edges of a regular simplex form a spherical two-distance set in R d .Recently Glazyrin and Yu [6] determined that the maximum size of spherical two-distance sets in R d is indeed 1  2 d(d + 1) whenever d ≥ 7 and d + 3 is not an odd perfect square; see [14,2,17] for results in many small dimensions.
Given a set A ⊂ [−1, 1), a spherical A-code is a set S of unit vectors in R d where x, y ∈ A for all distinct x, y ∈ S. We write N A (d) the maximum size of a spherical A-code in R d .In this paper, we are primarily interested in the case A = {α, β} for fixed −1 ≤ β < α < 1 and large d, in which case we write N α,β (d) instead of N {α,β} (d).
Jiang was supported by an AMS Simons Travel Grant and NSF Award DMS-1953946.Tidor was supported by the NSF Graduate Research Fellowship Program DGE-1745302.Yao and Zhang were supported by MIT UROP.Zhao was supported by NSF Award DMS-1764176, the MIT Solomon Buchsbaum Fund, and a Sloan Research Fellowship.
We recently solved Problem 1.1 in the case of equiangular lines [10] where β = −α.To state the result, we need the following spectral graph quantity, introduced in [9].Definition 1.2.The spectral radius order, denoted k(λ), of a real number λ > 0 is the smallest integer k so that there exists a k-vertex graph G whose spectral radius λ 1 (G) is exactly λ.Set k(λ) = ∞ if no such graph exists.(When we talk about the spectral radius or eigenvalues of a graph we always refer to its adjacency matrix.)Theorem 1.3 (Equiangular lines with a fixed angle [10]).Fix α ∈ (0, 1).Let λ = (1 − α)/(2α).For all sufficiently large d > d 0 (α), Let us recap some key steps in the proof of the upper bound on N α,−α (d) in Theorem 1.3.We will build on this framework.
Given a spherical {±α}-code S, we consider the associated graph G with vertex set S, where x, y ∈ S are adjacent in G if x • y = −α.We are allowed to replace any x ∈ S by −x without changing the equiangular lines configuration.An argument introduced in [1] reduces the problem to bounded degree graphs.
Theorem 1.4 ([1] and [10, Theorem 2.1]).For every α ∈ (0, 1), there exists ∆ depending only on α, such that given any spherical {±α}-code S in R n , one can replace some subset of vectors in S by their negations so that the associated graph G (as defined above) has maximum degree at most ∆.
The problem of bounding the size of S is related to the multiplicity of (1 − α)/(2α) as the second largest eigenvalue of the adjacency matrix of G.A crucial contribution of [10] is that every connected bounded degree graph has sublinear second eigenvalue multiplicity.More generally, we have the following.(See Definition 1.8 below for the precise definition of j-th eigenvalue multiplicity.)Theorem 1.5 ([10, Theorem 2.2]).For every j and ∆, there is a constant C = C(∆, j) so that every connected n-vertex graph with maximum degree at most ∆ has j-th eigenvalue multiplicity at most Cn/ log log n.
Turning to spherical two-distance sets, given a spherical {α, β}-code S (with β < 0 ≤ α as always throughout this paper), we define its associated graph G to have vertex set S and where x, y ∈ S are adjacent in G if x • y = β.Unlike for equiangular lines, here we are no longer allowed to negate a subset of vectors in a spherical {α, β}-code.Instead, we show that G is very close to a complete p-partite graph.Here p is a specific constant, with the equiangular lines problem corresponding to p = 2. Definition 1.6.A graph G is a ∆-modification of another graph H on the same vertex set if the symmetric difference of G and H has maximum degree at most ∆.
Remark.We allow empty parts in a complete p-partite graph.In particular, a complete t-partite graph is always a complete p-partite graph for t ≤ p.
It will be helpful to study such graphs using the language of signed graphs.Definition 1.8.A signed graph is a graph whose edges are each labeled by + or −.Throughout the paper we decorate variables for signed graphs with the ± superscript.The signed adjacency matrix A G ± of a signed graph G ± on n vertices is the n × n matrix whose (i, j)-th entry is 1 if ij is a positive edge, and −1 if ij is a negative edge, and 0 otherwise.We denote the eigenvalues of for the the multiplicity of λ as an eigenvalue of G ± .The j-th eigenvalue multiplicity of G ± is defined to be mult(λ j (G ± ), G ± ).We use |G| and |G ± | to denote the number of vertices in the graph.
Given a ∆-modification G of a complete p-partite graph K, we study the signed graph G ± defined by A G ± = A G − A K .The growth rate of N α,β (d) is related to the eigenvalue multiplicity of G ± .We introduce the following parameter generalizing the spectral radius order k(λ) for signed graphs.Definition 1.9.A valid p-coloring of a signed graph G ± is a coloring of the vertices using p colors such that the endpoints of every negative edge are colored using distinct colors, and the endpoints of every positive edge are colored using identical colors.(See Figure 1 for an example.)The chromatic number χ(G ± ) of a signed graph G ± is the smallest p for which G ± has a valid p-coloring.If G ± does not have a valid p-coloring for any p, we write χ(G ± ) = ∞.Definition 1.10.Given λ > 0 and p ∈ N, define the parameter We say that k p (λ) is achievable if it is finite and the infimum can be attained.
In the definition of k p (G ± ), it is enough to consider connected G ± , since the eigenvalues of G ± are given by the union of the the eigenvalues of its connected components.
If χ(G ± ) ≤ 2, then the signed graph G ± and its underlying graph G have the same eigenvalues (including multiplicities), since the signed adjacency matrix of G ± can be obtained by conjugating the adjacency matrix of G by a {±1}-valued diagonal matrix.By the Perron-Frobenius theorem, the top eigenvalue of a connected unsigned graph has multiplicity one.Thus, for all λ > 0. However the behavior of k p (λ) is far more mysterious when p ≥ 3. We do not know any general method of estimating or certifying values of k p (λ).Also, it is not even clear whether the infimum in the definition of k p (λ) can always be attained whenever k p (λ) is finite.Generalizing the construction in [9] relating equiangular lines to k(λ), we can obtain a lower bound on lim d→∞ N α,β (d)/d (see Proposition 2.2).Our main conjecture, below, says that this lower bound is sharp.
We see above that the parameters appear to play important roles in the problem.These two parameters λ and p conjecturally govern the asymptotic behavior of N α,β (d).Our main theorem below establishes Conjecture 1.11 for p ≤ 2, as well as for λ ∈ {1, √ 2, √ 3}.This is the first time that some lim d→∞ N α,β (d)/d is determined outside of the equiangular lines setting (α = −β).Theorem 1.12. .
Theorem 1.13.Fix −1 ≤ β < 0 ≤ α < 1. Set λ = (1 − α)/(α − β) and p = ⌊−α/β⌋ + 1 and q = max{1, p/2}.Then Our proof of Theorem 1.12 indeed confirms Conjecture 1.11 in all the solved cases, namely when We employ a number of different methods for bounding eigenvalue multiplicities in signed graphs in the different parts of Theorem 1.12: • For (a) and (b), we apply the sublinear bound on eigenvalue multiplicity of bounded degree unsigned graphs (Theorem 1.5 above; see Section 4).• For (c), we develop a forbidden induced subgraph framework (see Section 5), and we apply a careful third moment and triangle counting argument (see Section 6).• For (d) we apply an algebraic degree argument (see Section 7).Additionally, we confirm Conjecture 1.11 for all algebraic integers λ whose degree equals k(λ) (see the end of Section 7).
Remark.After this work is completed, building on our forbidden induced subgraph framework, Jiang and Polyanskii [8] proved Conjecture 1.11 for every λ < λ * , where λ * = β 1/2 + β −1/2 ≈ 2.01980 and β is the unique real root of A major obstacle to completely settling Conjecture 1.11 is that bounded degree signed graphs may have linear top eigenvalue multiplicity.
Theorem 1.14.For every n ≥ 3, there is a connected signed graph with 6n vertices, maximum degree 5, and chromatic number 3, such that its largest eigenvalue appears with multiplicity n.
The rest of the paper is organized as follows.In Section 2, we explain the connection with spherical two-distance sets and the spectral theory of signed graphs, and further proves a lower bound on N α,β (d).In Section 3 we prove the structural result, Theorem 1.7.In Section 4 we prove Theorem 1.12(a), Theorem 1.12(b), and Theorem 1.13 using Theorem 1.5.In Section 5 we develop a forbidden induced subgraph framework to bound N α,β (d) from above.In Section 6 we prove Theorem 1.12(c) via a third moment argument under the forbidden induced subgraph framework.In Section 7 we prove Theorem 1.12(d) via an algebraic argument.In Section 8 we give two constructions related to Theorem 1.14.

Connection to spectral theory of signed graphs
The spherical two-distance set problem has the following equivalent spectral graph theoretic formulation.Here A 0 means that A is positive semidefinite.
There exists a spherical {α, β}-code of size N in R d if and only if there exists a graph G on N vertices satisfying Proof.For a spherical {α, β}-code {v 1 , . . ., v N } in R d , let G be the associated graph on vertex set {1, . . ., N }, where ij is an edge whenever v i , v j = β.The Gram matrix M = ( v i , v j ) i,j has 1's on its diagonal and α, β everywhere else, so it equals We are now ready to establish a lower bound on N α,β (d) using Lemma 2.1.
Proof.Let µ = α/(α − β).Take G to be d-vertex graph with no edges, so that A G = 0 and λI − A G + µJ is positive semidefinite and has rank at most d.So N α,β (d) ≥ d by Lemma 2.1.In fact, the spherical two-distance set constructed here forms a regular (d − 1)-simplex.
Hereafter assume that k p (λ) < ∞.We first construct, for every signed graph . ., V p be the color classes of a valid p-coloring.Consider the unsigned graph G obtained from taking the symmetric difference between the underlying graph of G ± and the complete p-partite graph with parts V 1 , . . ., V p .The adjacency matrix of G is related to the signed adjacency matrix of G ± by where K is the complete p-partite graph with parts V 1 , . . ., V p .Therefore, We now note that µJ − A K is positive semidefinite.Indeed, for every x ∈ R V (G ± ) , we set s i = v∈V i x v for each i ∈ {1, . . ., p}, and we see Therefore µJ − A K 0, and so λI − A G + µJ 0. We conclude by Lemma 2.1 that there exists a spherical {α, β}-code of size Now fix an arbitrary ε > 0. Take a signed graph Thus we can apply the above construction to .
Finally notice that when k p (λ) is achievable, we can take ε = 0 in the above argument.

Structure of the associated graph
In this section we prove Theorem 1.7, which gives a structure characterization of graphs that can arise from a spherical two-distance set.To that end, we introduce the following notation.
to be the set of vertices in V (G) \ X that are adjacent to all vertices in Y and not adjacent to any vertices in X \ Y , and for a set X ⊆ V (G) and ∆ ∈ N, define We now present a series of structural lemmas leading to the proof of Theorem 1.7.
(a) Neither of the following is an induced subgraph of G: (a1) the complete graph K ∆ ; (a2) the complete (p + 1)-partite graph K ∆,...,∆ , where Proof of (a1).Suppose on the contrary that G contains K ∆ as a subgraph.Let v ∈ R V (G) be the vector that assigns 1 to vertices in K ∆ and 0 otherwise.Then v ⊺ (λI which would be negative if we had chosen ∆ > (1 + λ)/(1 − µ).
Proof of (b1).Suppose on the contrary that a vertex u ∈ C X,∆ has ∆ neighbors v 1 , . . ., v ∆ ∈ C X,∆ .Let v ∈ R V (G) be the vector that assigns L to u, λL/∆ to v 1 , . . ., v ∆ , −(λ + 1) to the vertices in X, and 0 otherwise.Because u, v 1 , . . ., v ∆ ∈ C X,∆ , we have Using this bound and the fact that v ⊺ 1 = 0, we obtain that v ⊺ (λI which would be negative for sufficiently large L if we had chosen ∆ > λ 2 .

Proof of (b2). To show that |V
For any α, β, γ ∈ R, we consider the vector that assigns α to the vertices in A, β to the vertices in X \ A, γ to the vertices in C X (A), and 0 otherwise, and we have In particular, taking β = −(aα + cγ)/(b + λ/µ), we obtain that for all α, γ ∈ R, For this quadratic form in α and γ to be positive semidefinite, its discriminant must be nonpositive: By the assumption that a, b > ∆, if we had taken ∆ ≥ max{λ/µ, 4λ 2 , 2}, then λ < µb and λ 2 < b/4, hence (1) would imply the following series of inequalities: Proof of (c1).Suppose on the contrary that a vertex ) be the vector that assigns L to v, −λL/∆ to v 1 , . . ., v ∆ , −1 to the vertices in X 1 , λ to the vertices in X 2 , and Using this bound and the fact that v ⊺ 1 = 0, we obtain that v ⊺ (λI which would be negative for sufficiently large L if we had chosen ∆ > λ 2 .
Choose ∆ and L 0 as in Lemma 3.2.We shall prove that G, after removing at most pL2 L + p 2 ∆ + R(∆, L2 pL ) vertices, is a p∆-modification of a complete p-partite graph, where L = L 0 + (p + 2)∆ and R(•, •) is the Ramsey number.
We may assume that |G| ≥ R(∆, L) because otherwise G is vacuously a p∆-modification of a complete p-partite graph after removing all its vertices.By Lemma 3.2(a1) and Ramsey's theorem, there exists an independent set of size L in G. Choose the maximum t ≤ p such that the complete t-partite graph K L−t∆,...,L−t∆ is an induced subgraph of G (note that t ≥ 1 since there is an independent set of size L).Let X 1 , . . ., X t ⊂ V (G) be the parts of this t-partite graph.
Define for every i ∈ {1, . . ., t} the vertex subset By (b1) and (c1) in Lemma 3.2, we see that the Finally, we claim that U + does not contain a subset of size L2 tL that is independent in G. Indeed, suppose on the contrary that U + contains an independent set of size L2 tL .Since every vertex in U + has at least L − (t + 1)∆ neighbors in X i for each i, by the pigeonhole principle, there exist ] is a complete (t + 1)-partite graph with parts X ′ 1 , . . ., X ′ t and U ′ , which contradicts our choice of t or Lemma 3.2(a2) in case t = p.This finishes the proof of the claim.In view of Lemma 3.2(a1) and Ramsey's theorem, we obtain
For each i ≤ s, since G i is a connected graph with maximum degree at most ∆ and λ j (G i ) ≤ λ, Theorem 1.5 gives a constant C = C(∆, j) such that We break the rest of the proof into two cases.
For each i > s, when λ 1 (G i ) = λ, because G i is connected, we know that n i ≥ k(λ), and so by the Perron-Frobenius theorem, we obtain when λ 1 (G i ) < λ, clearly (4) holds trivially.We combine (3) and ( 4) to obtain .

Now the signed adjacency matrix of G
Note that rank(µJ −A K ) ≤ p. From the first condition above, we deduce using the Courant-Fischer theorem that λ p+1 (λI − A G ± ) ≥ 0 or equivalently λ p+1 (G ± ) ≤ λ.From the second condition above, we deduce using subadditivity of matrix ranks that rank(λI We break the rest of the proof into two cases.
Case p = 1.The signed graph G ± consists of positive edges only.Lemma 4.1 provides the upper bound Combining with (5), we get which implies The desired upper bound on N α,β (d) follows immediately in view of Case p ≥ 2. Let V 1 and V 2 be the largest parts of the complete p-partite graph K. Let G ± 12 be the signed subgraph of G ± induced on V 1 ∪ V 2 , and let G 12 be the underlying graph of G ± 12 .Notice that By the Cauchy interlacing theorem, we have ). Combining (5) and the above two inequalities, we get which implies The desired upper bound on N α,β (d) follows immediately in view of the inequalities As a corollary, we obtain the following general lower bound on k p (λ).
Proof.Comparing Proposition 2.2 and Theorem 1.13, we get , which implies the desired lower bound.(It is also not hard to prove Corollary 4.2 directly, but we do not do so here.) Remark.For general λ, we do not know any algorithm for computing k(λ) (or even deciding whether k(λ) < ∞), though deciding whether k(λ) < k for each integer k is a finite problem as can be done by a brute-force search over all graphs up to a fixed size.When λ ∈ N, we have k(λ) = λ + 1 because the complete graph K λ+1 is the graph on fewest vertices with spectral radius λ.In contrast, even for λ ∈ N, computing the exact values of k p (λ) seems to be very difficult for p ≥ 3.For λ = 2, Corollary 4.2 implies that k 3 (2) ≥ 9/5 and k 4 (2) ≥ 3/2.Note that both the Paley graph of order 9 in Figure 2 and the Shrikhande graph in Figure 3 are strongly regular graphs with −2 as their smallest eigenvalue with multiplicity 4 and 9 respectively.Moreover their chromatic numbers are 3 and 4 respectively.The all-negative signed graphs of these two strongly regular graphs would yield k 3 (2) ≤ 9/4 and k 4 (2) ≤ 16/9.We leave the determination of k p (2) for p ≥ 3 as an open problem.Theorem 1.12(a) and Theorem 1.12(b) follow easily from Theorem 1.13 and Proposition 2.2.
Corollary 4.2 implies that k p (1) ≥ p/(p − 1).To see that p/(p − 1) can be achieved for k p (1), consider the all-negative complete signed graph K ± p on p vertices.Clearly χ(K ± p ) = p.Since the smallest eigenvalue of the complete unsigned graph K p is −1 with multiplicity p − 1, the largest eigenvalue of K ± p is 1 with multiplicity p − 1.Now Proposition 2.2 provides a lower bound that matches (6) up to an additive constant.

Forbidden induced subgraphs
The next lemma enables us to forbid finitely many induced subgraphs in the signed graph that arises from Theorem 1.7.Here an induced subgraph of a signed graph keeps the original edge signs.Lemma 5.1.Fix λ > 0, µ ∈ (0, 1), p ∈ N, and ∆ ∈ N.For every signed graph H ± with λ 1 (H ± ) > λ, there exists n 0 ∈ N such that for every t ≤ p and every graph G that is a ∆-modification of a complete t-partite graph K, if λI − A G + µJ 0, and the size of each part of K is at least n 0 , then H ± cannot be an induced subgraph of the signed graph G ± defined by Proof.Suppose that G is a ∆-modification of a complete t-partite graph K with parts V 1 , . . ., V t , and suppose that the size of each part of K is at least n 0 .Assume for the sake of contradiction that G induces a complete t-partite graph with parts V 1 , . . ., V t , (3) for every vertex v of H ± , if v ∈ V i , then, in G, the vertex v is adjacent to every vertex in V j for j = i, and is not adjacent to any vertex in V i .
Let x ∈ R V (H ± ) be a top eigenvector of H ± , and set Consider the vector v ∈ R V (G) extending x that in addition assigns −s i /m to each vertex in V i for i ∈ {1, . . ., t}.Since v is chosen so that u∈ V i v u = 0 for each i ∈ {1, . . ., t}, we have Jv = 0 and A K v = 0. Now we can simplify the quadratic form as follows: which is negative because m > λ|H ± |/(λ 1 (H ± ) − λ).This contradicts λI − A G + µJ 0.
Lemma 5.1 leads us to bound eigenvalue multiplicities in a restricted class of signed graphs obtained by forbidding certain induced subgraphs.Definition 5.2.Given a family H of signed graphs, let M p,H (λ, N ) be the maximum possible value of mult(λ, G ± ) over all signed graphs G ± on at most N vertices that do not contain any member of H as an induced subgraph and satisfy χ(G ± ) ≤ p and λ p+1 (G ± ) ≤ λ.
In our application, we will only be allowed to forbid a finite H such that λ 1 (H ± ) > λ for all H ± ∈ H.
Remark.We could choose H properly so that every signed graph G ± considered in Definition 5.2 of M p,H (λ, N ) has its maximum degree bounded by a constant depending only on p and λ.In fact, set D = ⌊λ 2 ⌋, and suppose that H includes all the signed graphs H ± on D + 2 vertices with χ(H ± ) ≤ 2 such that the underlying graph of H ± contains the star K 1,D+1 .One can then show that for every graph G ± that does not contain any member of H as an induced subgraph, the maximum degree of G ± is at most χ(G ± )D.
The next statement relates the maximum size of a spherical two-distance set with the above eigenvalue multiplicity quantity.Let n 0 = n 0 (α, β, H) be the maximum n 0 given by Lemma 5.1 when it is applied to each member of H respectively with the parameters λ, µ, p, and ∆.After removing at most ∆ vertices from G, we can further remove at most pn 0 vertices from G to obtain a graph, denoted G, that is a ∆-modification of a t-partite graph, denoted K, with each part of size at least n 0 , for some t ≤ p. Define the signed graph G ± by A G ± = A G − A K .Since λI − A G + µJ 0, by our choice of n 0 , we know that the signed graph G ± does not contain any member of H as an induced subgraph.Notice that χ(G ± ) ≤ t ≤ p.Now the signed adjacency matrix of G ± satisfies Note that rank(µJ − A K ) ≤ t ≤ p. From (8a) we deduce using the Courant-Fischer theorem that λ p+1 (λI − A G ± ) ≥ 0 or equivalently λ p+1 (G ± ) ≤ λ.Recall that G ± has at most N α,β (d) vertices, G ± does not contain any member of H as an induced subgraph, and χ(G ± ) ≤ p.According to Definition 5.2, From (8b) we deduce using subadditivity of matrix ranks that rank(λI Combining with .
For each value of λ and p, if we could prove the following upper bound on the eigenvalue multiplicity, then it would imply Conjecture 1.11 via Theorem 5.3.
Proof of Claim 2. Let w ∈ V (G) \ {v 0 , v 1 , v 2 , v 3 } be a vertex that is adjacent to at least one of v 0 , v 1 , v 2 , v 3 , and consider the vector v ∈ R W , where W = {v 0 , v 1 , v 2 , v 3 , w}, that assigns √ 3 to v 0 , σ(v 0 v i ) to v i for i ∈ {1, 2, 3}, ε to w, where σ : E(G) → {±1} is the signing of G ± and ε ∈ R. According to our choice of v, we have By the Courant-Fischer theorem, we also have For the last inequality to hold for all ε ∈ R, we must have which implies that v 0 w ∈ E(G), and exactly two of v 1 , v 2 , v 3 are adjacent to w in G. ♦ Claim 3. The maximum degree of G is at most 4.
Proof of Claim 3. Suppose on the contrary that v 0 is adjacent to at least 5 vertices in G. Without loss of generality we may assume that v 0 ∈ V 1 , and by the pigeonhole principle that 3 neighbors, say by Claim 1, H ± contains no triangles.Thus G induces a star on {v 0 , v 1 , v 2 , v 3 } centered at v 0 , and so by Claim 2, v 0 has no neighbors other than v 1 , v 2 , v 3 in G, which leads to a contradiction.♦ Claim 4. The underlying graph G contains an induced star K 1,3 .
Proof of Claim 4. Suppose on the contrary that G does not contain any induced K 1,3 .For every v ∈ V (G), the subgraph of G induced by the neighbors of v contains no independent set of size 3, in particular, this induced subgraph contains at most 2 connected components, hence it contains at least d v − 2 edges, where d v is the degree of v in G.In other words, every v ∈ V (G) is contained in at least d v − 2 triangles.Recall from Claim 2 that every triangle in G has all its edges negatively signed.Let λ 1 , λ 2 , . . ., λ n be the eigenvalues of G ± , where n = |G ± |, and let t be the total number of triangles in G. Thus we have

Thus we have
Since the characteristic polynomial of A G ± is a polynomial with integer coefficients, we obtain mult(− √ 3, G ± ) = mult( √ 3, G ± ), which is more than 3n/7.For other eigenvalues λ i , by Claim 3, we know that λ i ≥ −4, and so Mowshowitz [13] states that such a graph must be asymmetric 2 .Asymmetric graphs have at least 6 vertices.There are 8 such graphs on 6 vertices [5].Among these 8 asymmetric graphs on 6 vertices, exactly 7 of them have irreducible characteristic polynomials, 3 hence their spectral radii satisfy k(λ) = deg(λ). .

Signed graphs with large eigenvalue multiplicities
In contrast to Theorem 1.5, there exist connected signed graphs with bounded maximum degree and chromatic number and linear largest eigenvalue multiplicity.In this section, we show two such constructions.These constructions illustrate an important obstacle to proving Conjecture 1.11 following the current framework introduced in [10].
Example 8.1.Let n ≥ 3. Let G ± n be the signed graph consisting of (see Figure 8 for an illustration of G n copies of a signed K 5 with 3 positive edges forming a K 3 , and (3) for each i ∈ {1, . . ., n}, a positive edge connecting v i and u + i , a negative edge connecting v i and u − i , where u + i and u − i are the two vertices outside the positive K 3 in the i-th copy of K 5 .
2 An asymmetric graph is a graph for which there are no automorphisms other than the trivial one. 3It was asserted in [9,Section 4] that all 8 asymmetric graphs on 6 vertices have irreducible characteristic polynomials.However the characteristic polynomial of the asymmetric graph is x(x 5 − 8x 3 − 6x 2 + 8x + 6).
So G ± n is a signed graph on 6n vertices of maximum degree 5 and chromatic number 3. However the multiplicity of its largest eigenvalue is linear in |G ± n |.Theorem 1.14 is an immediate consequence of the following result.Proposition 8.2.The largest eigenvalue of G ± n is ( √ 33 + 1)/2 with multiplicity n.
Proof.We denote by K ± 5 the signed K 5 with 3 positive edges forming a K 3 , and we compute the spectrum of K ± 5 to be (1− √ 33)/2, −1, −1, 1, (1+ √ 33)/2.Because the largest eigenvalue ( √ 33+1)/2 is simple, by symmetry the corresponding eigenvector assigns the same value to u + i and u − i .For the i-th copy of K ± 5 in G ± n , we can extend its top eigenvector to a vector x i on V (G ± n ) by padding zeros.Since where A denotes the signed adjacency matrix of G ± n , the vector x i is also an eigenvector of G ± n associated with the eigenvalue ( √ 33 + 1)/2.For every vector x ∈ R V (G ± n ) that is perpendicular to all x i , 1 ≤ i ≤ n, we claim that x ⊺ Ax ≤ 3x ⊺ x, and so all the eigenvalues other than the ones corresponding to x 1 , . . ., x n are at most 3. Take such a vector x, and set U = {u + 1 , u − 1 , . . ., u + n , u − n } and V = {v 1 , . . ., v n }.We take the orthogonal decomposition x = y + z such that y and z are supported respectively on V (G ± n ) \ V and U ∪ V .In particular, for every i ∈ {1, . . ., n}, One can check that y ⊺ Az = 0. We can simplify x ⊺ Ax = (y + z) ⊺ A(y + z) = y ⊺ Ay + z ⊺ Az.
Since x and z are both orthogonal to each x i , so is y = x − z.By the Courant-Fischer theorem, we obtain y ⊺ Ay ≤ λ 2 (K ± 5 )y ⊺ y = y ⊺ y.As z is supported on U ∪ V , we bound z ⊺ Az by bounding the spectral radius of G ± n [U ∪ V ].Since the chromatic number of G ± n [U ∪ V ] is 2, the induced signed subgraph shares the same spectral radius with its underlying graph, denoted H, on U ∪ V .Notice that the vector that assigns 1 to U and 2 to V is an eigenvector of H with positive components associated with the eigenvalue 3.By the Perron-Frobenius theorem, the spectral radius of H is 3. Thus z ⊺ Az ≤ 3z ⊺ z.Recall that x = y + z is an orthogonal decomposition.Thus xAx = y ⊺ Ay + z ⊺ Az ≤ y ⊺ y + 3z ⊺ z ≤ 3(y ⊺ y + z ⊺ z) = 3x ⊺ x.
Even if we restrict the signed graph G ± to be all-negative, its largest eigenvalue multiplicity could still be linear in |G ± |.It suffices to construct the underlying graph G with bounded maximum degree whose smallest eigenvalue multiplicity is linear in |G|.
Example 8.3.Let n ≥ 3. Let H n be the (unsigned) graph consisting of (see Figure 9 for an illustration of H 6 ) (1) an n-cycle on v 1 , v 2 , . . ., v n , (2) n copies of K 3,3 , and (3) for each i ∈ {1, . . ., n}, two edges connecting v i to u 1 i and u 2 i , where u 1 i and u 2 i are two adjacent vertices in the i-th copy of K 3,3 .So H n is a graph on 7n vertices of maximum degree 4.Moreover, since the chromatic number of H n is 3, the corresponding all-negative signed graph has the same chromatic number.
Proposition 8.4.The smallest eigenvalue of H n is −3 with multiplicity n.

Figure 1 .
Figure 1.A valid 3-coloring of a signed graph.Throughout this paper, the positive edges are represented by solid segments and the negative edges are represented by dashed segments.
αJ, where I is the identity matrix, J the all-ones matrix, and A G the adjacency matrix of G.We have M/(α−β) = λI−A G +µJ, where λ = (1 − α)/(α − β) and µ = α/(α − β).Since the Gram matrix M is positive semidefinite and has rank at most d, the same holds for λI − A G + µJ.Conversely, for every G, λ and µ for which λI − A G + µJ is positive semidefinite and has rank d, there exists a corresponding configuration of N unit vectors in R d , with pairwise inner products in {α, β}.