Equiangular lines and large multiplicity of fixed second eigenvalue

TL;DR

This paper demonstrates the existence of infinitely many angles where the maximum number of lines meeting at the origin in ^n exceeds a linear function of n by a logarithmic factor, using novel graph constructions.

Contribution

It constructs regular graphs with fixed second eigenvalue and large multiplicity, advancing understanding of equiangular lines and eigenvalue multiplicities.

Findings

Existence of infinitely many angles with high line configurations

Construction of regular graphs with prescribed eigenvalues and large multiplicity

New distribution method on bipartite graph factors

Abstract

Answering a question of Jiang and Polyanskii as well as Jiang, Tidor, Yao, Zhang, and Zhao, we show the existence of infinitely many angles $\theta$ for which the maximum number of lines in $\mathbb R^n$ meeting at the origin with pairwise angles $\theta$ exceeds $n+\Omega(\log\log n)$ but is at most $n+o(n)$. To accomplish this, we construct, for various real $\lambda$ and integer $d$, $d$-regular graphs with second eigenvalue exactly $\lambda$ and arbitrarily large second eigenvalue multiplicity. Central to our construction is a distribution on factors of bipartite graphs which possesses concentration properties.

Equations

$$n+\Omega (\log \log n)\le N_{\alpha }(n)\le n+O(n/\log \log n).$$

$$A_{\tilde{G}}=\frac{1}{2}\left(\begin{array}{cc}{A}_G+A_s& A_G-A_s\\ A_G-A_s& A_G+A_s\end{array}\right).$$

$$\mathbb{E}[A_s]=\mathbb{E}[A_G-2A_H]=\left(1-\frac{2a}{d}\right)A_G.$$

$$\mathrm{Pr}\left[\left|u^{⊺}\left(A_H-\frac{a}{d}{A}_G+M\right)v\right|>t(2+\sqrt{2})\sqrt{dn\lceil {\log }_{2}n\rceil }\Vert u{\Vert }_{\infty }\Vert v{\Vert }_{\infty }\right]\le 2({\log }_{2}d)e^{-t^2/2}.$$

$$\mathrm{tr}{A}_G^{2r}=\sum _{i=1}^n{\lambda }_i(A_G{)}^{2r}\ge {\lambda }_1(A_G{)}^{2r}.$$

$$n(4d{)}^r\ge {\lambda }_1(A_G{)}^{2r},$$

$${\lambda }_1(A_G)\le n^{1/(2r)}\sqrt{4d}\le n^{\frac{1}{2{\log }_{2}n}}\sqrt{4d}=(2\sqrt{2})\sqrt{d}.$$

$$\mathrm{Pr}\left[\left|u^{⊺}\left(A_H-\frac{1}{2}{A}_G+M\right)v\right|\ge t\sqrt{dn\lceil {\log }_{2}n\rceil }\Vert u{\Vert }_{\infty }\Vert v{\Vert }_{\infty }\right]\le 2e^{-t^2/2}.$$

$$A_H-\frac{a}{d}{A}_G+M=A_{H_0}+A_{H_1}-\frac{1}{2}{A}_{G_0}-\frac{1}{2}{A}_{G_1}+\frac{1}{2}{A}_{G_0}-A_{H_0}=A_{H_1}-\frac{1}{2}{A}_{G_1}.$$

$$X=u^{⊺}\left(A_{H_1}-\frac{1}{2}{A}_{G_1}\right)v.$$

$$X=\sum _{C\in \mathcal{C}}{X}_C{u}^{⊺}\left(\frac{1}{2}{\tilde{A}}_C\right)v,$$

$$\left|u^{⊺}\left(\frac{1}{2}{\tilde{A}}_C\right)v\right|\le \mathrm{len}(C)\Vert u{\Vert }_{\infty }\Vert v{\Vert }_{\infty }.$$

$$\mathrm{Pr}\left[|X|\ge \mu \right]\le 2\exp \left(\frac{-2{\mu }^2}{{\sum }_{C\in \mathcal{C}}{\left(2\mathrm{len}(C)\Vert u{\Vert }_{\infty }\Vert v{\Vert }_{\infty }\right)}^2}\right).$$

$$\sum _{C\in \mathcal{C}}\mathrm{len}(C{)}^2\le \left(\underset{C\in \mathcal{C}}{\max }\mathrm{len}(C)\right)\sum _{C\in \mathcal{C}}\mathrm{len}(C)\le 2\lceil {\log }_{2}n\rceil E(G)=dn\lceil {\log }_{2}n\rceil ,$$

$$\mathrm{Pr}\left[|X|\ge \mu \right]\le 2\exp \left(\frac{-{\mu }^2}{2dn\lceil {\log }_{2}n\rceil \Vert u{\Vert }_{\infty }^2\Vert v{\Vert }_{\infty }^2}\right).$$

$$\mathrm{Pr}\left[\left|u^{⊺}\left(A_H-\frac{a}{d}{A}_G+M\right)v\right|>t(2+\sqrt{2})\sqrt{dn\lceil {\log }_{2}n\rceil }\Vert u{\Vert }_{\infty }\Vert v{\Vert }_{\infty }\right]\le 2({\log }_{2}d)e^{-t^2/2}.$$

$$\Vert M\Vert \le \Vert M_2\Vert +(2a/d)\Vert M_1\Vert \le c_{d/2}+1\cdot \sqrt{2d}=c_d$$

$$A_H-\frac{a}{d}{A}_G+M=\left(A_H-\frac{a}{d/2}{A}_{G_1}+M_2\right)+\frac{2a}{d}\left(A_{G_1}-\frac{1}{2}{A}_G+M_1\right),$$

$$\mathrm{Pr}\left[\left|u^{⊺}\left(A_{G_1}-\frac{1}{2}{A}_G+M_1\right)v\right|\ge t\sqrt{dn\lceil {\log }_{2}n\rceil }\Vert u{\Vert }_{\infty }\Vert v{\Vert }_{\infty }\right]$$

$$\mathrm{Pr}\left[\begin{array}{c}\left|u^{⊺}\left(A_H-\frac{a}{d/2}{A}_{G_1}+M_2\right)v\right|\\ \ge t(2+\sqrt{2})\sqrt{(d/2)n\lceil {\log }_{2}n\rceil }\Vert u{\Vert }_{\infty }\Vert v{\Vert }_{\infty }\end{array}\right]$$

$$t(2+\sqrt{2})\sqrt{(d/2)n\lceil {\log }_{2}n\rceil }\Vert u{\Vert }_{\infty }\Vert v{\Vert }_{\infty }+\frac{2a}{d}\left(t\sqrt{dn\lceil {\log }_{2}n\rceil }\Vert u{\Vert }_{\infty }\Vert v{\Vert }_{\infty }\right)$$

$$(2+\sqrt{2})\sqrt{d/2}+\frac{2a}{d}\sqrt{d}\le (\sqrt{2}+1)\sqrt{d}+\sqrt{d}=(2+\sqrt{2})\sqrt{d},$$

$$A_H-\frac{a}{d}{A}_G+M=(A_G-A_{H^{\prime }})-\frac{a}{d}{A}_G-M^{\prime }=-\left(A_{H^{\prime }}-\frac{d-a}{d}{A}_G+M^{\prime }\right),$$

$$A_H-\frac{a}{d}{A}_G+M$$

$$=A_H-\frac{a}{d}{A}_{G_1}-\frac{a}{d(d-1)}{A}_{G_1}+M_1$$

$$=A_H-\frac{a}{d-1}{A}_{G_1}+M_1,$$

$$\Vert M\Vert \le \frac{a}{d}\Vert A_{H^{\prime }}\Vert +\frac{a}{d(d-1)}\Vert A_{G_1}\Vert +\Vert M_1\Vert \le \frac{a}{d}+\frac{a}{d}+c_{d-1}\le c_{d-1}+1$$

$$c_d$$

$$\le (2+\sqrt{2})(\sqrt{2d}-1)+{\log }_{2}d$$

$${\log }_{2}d-(2+\sqrt{2})\le (4-2\sqrt{2})\sqrt{d},$$

$$\left|v^{⊺}\left(A_H-\frac{a}{d}{A}_G\right)v\right|\le 7\sqrt{d}$$

$$2({\log }_{2}d)e^{-t^2/2}\le m{e}^{-3.25\ln m}<m^{-2}.$$

Full text

1. Introduction

A set of lines in $\mathbb{R}^{n}$ passing through the origin is called equiangular if each pair meets at the same angle $\theta$. The question of the maximum number of equiangular lines in given dimension is natural and well-studied. A short linear-algebraic argument due to Gerzon (see [LS73]) gives an upper bound in dimension $n$ of $\binom{n+1}{2}$, and a lower bound on the order of $n^{2}$ holds due to a construction of de Caen [dC00].

One may also consider the setting where the common angle is fixed. For an $\alpha\in(0,1)$, let $N_{\alpha}(n)$ denote the maximum number of lines in $\mathbb{R}^{n}$ each pair of which meets at an angle $\arccos\alpha$. One can see that $N_{\alpha}(n)\geq n$ for each $\alpha$ and $n$ by constructing the lines spanned by vectors generating the Gram matrix with ones on the diagonal and $\alpha$ off-diagonal. On the other hand, $N_{\alpha}(n)\leq n+2$ unless $(1-\alpha)/(2\alpha)$ is a totally real algebraic integer which exceeds all of its Galois conjugates [JP20, Proposition 23].

Much of the literature on $N_{\alpha}(n)$ focuses on the limit $\lim_{n\to\infty}N_{\alpha}(n)/n$. Due to a recent result of Jiang, Tidor, Yao, Zhang, and Zhao [JTY*+*21] confirming a conjecture of Jiang and Polyanskii [JP20, Conjecture A], this limit has (essentially) been completely determined. We begin by stating their result, for which one definition is necessary. Whenever we speak of the eigenvalues of a graph $G$, we refer to the eigenvalues of its adjacency matrix $A_{G}$.

Definition 1.1**.**

For a positive real $\lambda$, the spectral radius order $k(\lambda)$ of $\lambda$ is the minimum number of vertices of a graph with largest eigenvalue exactly $\lambda$, or $\infty$ if no such graph exists.

Theorem 1.2** **([JTY*+*21, Theorem

1.2]).

Fix $\alpha\in(0,1)$, and let $\lambda=(1-\alpha)/(2\alpha)$.

1. (a)

If $k=k(\lambda)$ is finite, then $N_{\alpha}(n)=\lfloor k(n-1)/(k-1)\rfloor$ for all sufficiently large $n$. 2. (b)

If $k(\lambda)=\infty$, then $N_{\alpha}(n)=n+o(n)$.

The $o(n)$ term in case (b) given by [JTY*+*21] is on the order of $n/\log\log n$, and arises from the same bound on the multiplicity of the second largest eigenvalue of the adjacency matrix of a bounded degree graph. This result inspired investigation into bounds on this multiplicity. McKenzie, Rasmussen, and Srivastava [MRS21] improved the upper bound to $n/\log^{1/5-o(1)}n$ in the case of regular graphs, and also provided sublinear upper bounds in the unbounded degree case. On the other side, Haiman, Schildkraut, Zhang, and Zhao [HSZZ22] gave a lower bound (i.e. a construction of bounded-degree graphs with large second eigenvalue multiplicity) of $n^{1/2-o(1)}$.

While Theorem 1.2 determines the main term of $N_{\alpha}(n)$ as $n$ grows, it does not establish the order of secondary terms in the $k(\lambda)=\infty$ case. The following bound is conjectured in [JP20] and [JTY*+*21]:

Conjecture 1.3** ([JP20, Conjecture B], [JTY*+*21, Conjecture 6.1]).**

Fix $\alpha\in(0,1)$, and let $\lambda=(1-\alpha)/(2\alpha)$. If $k(\lambda)=\infty$, then $N_{\alpha}(n)=n+O_{\alpha}(1)$.

Our main result is a disproof of 1.3 for infinitely many $\alpha$, demonstrating the existence of “intermediate” behavior of $N_{\alpha}(n)$.

Theorem 1.4**.**

For infinitely many $\alpha\in(0,1)$, the function $N_{\alpha}(n)$ satisfies

[TABLE]

We obtain this result via constructing a family of bounded-degree regular graphs with second eigenvalue exactly $\lambda=(1-\alpha)/(2\alpha)$ for various $\lambda$. The second eigenvalue multiplicity we construct is much smaller than that in [HSZZ22]. However, to our knowledge, this is the first result giving unbounded second eigenvalue multiplicity when the eigenvalue is fixed.

Theorem 1.5**.**

For infinitely many positive real $\lambda$, there exist infinitely many connected $n$-vertex graphs with bounded degree and second eigenvalue exactly $\lambda$ of multiplicity $\Omega(\log\log n)$. In particular, this holds when

1. (a)

$\lambda$* is a sufficiently large integer, or* 2. (b)

$\lambda=2\sqrt{u^{2}+1}-1$* for a sufficiently large integer $u$.*

In fact, we can prove this result for any real $\lambda$ which is the largest eigenvalue of a symmetric integer matrix satisfying some technical conditions; see Definition 5.3 and Corollary 5.6. A more precise version of part (b) — Theorem 5.8 — will be enough to give Theorem 1.4.

2. Proof overview

In this section, we give an overview of how the graphs satisfying Theorem 1.5 are constructed. Inspired by Marcus, Spielman, and Srivastava [MSS15], the construction will proceed inductively via the $2$-lift operation, first studied in conjunction with eigenvalues by Bilu and Linial [BL06]. We construct a sequence $G_{0},G_{1},\dots$ of graphs so that each $G_{i}$ has only one eigenvalue greater than $\lambda$ and for which $\lambda$ is an eigenvalue of $G_{i}$ of multiplicity at least $i$.

For any graph $G$, we denote by $A_{G}$ its adjacency matrix, the $V(G)\times V(G)$ matrix where the entry $vw$ for $v,w\in V(G)$ is $1$ if and only if $vw\in E(G)$. For any square matrix $M$ and any positive integer $i$ at most the dimension of $M$, we let $\lambda_{i}(M)$ denote the $i$th largest eigenvalue of $M$, where eigenvalues are counted with multiplicity.

Integer second eigenvalue

When $\lambda$ is an integer, i.e. to obtain Theorem 1.5(a), the process is as follows:

1. (1)

Begin with the bipartite graph $G_{0}=K_{d,d}$ for some $d$ slightly larger than $\lambda$. 2. (2)

From $G_{i}$, repeatedly apply $2$-lifts which (i) add no new large eigenvalues and (ii) decrease the $L^{\infty}$ norm of the existing eigenvectors with large eigenvalue. To construct such lifts, we use a result of Marcus, Spielman, and Srivastava [MSS15]; the precise properties we need are stated here in Corollary 2.2. Call the graph resulting from these successive lifts $G_{i}^{\prime}$. 3. (3)

Once we have performed these lifts, select a subset of edges of $G_{i}^{\prime}$ randomly from an appropriate distribution (see Section 3) such that each vertex of $G_{i}^{\prime}$ is incident to exactly $a:=(d-\lambda)/2$ edges in the subset. This subset informs a choice of $2$-lift $G_{i+1}$ of $G_{i}^{\prime}$. For this lift, the vector assigning $1$ to all vertices in one copy of $V(G_{i}^{\prime})$ and $-1$ to all vertices in the other copy is an eigenvector of $A_{G_{i+1}}$ with eigenvalue $d-2a=\lambda$. Assuming $d$ is chosen suitably, this is with positive probability the largest new eigenvalue added.

Non-integer second eigenvalue

To obtain Theorem 1.5(b), we follow the same general strategy as above, but with some modifications. In step (1), instead of constructing $d$-regular bipartite graphs, we use $k$-partite graphs for which the graph between each pair of parts is a regular bipartite graph. In step (2), we perform the $2$-lifts of Marcus, Spielman, and Srivastava on each pair of parts. Finally, in (3), we choose a subset of edges from between each pair of parts which forms a regular bipartite graph of specified degree. Varying the degrees we choose between parts allows us to vary the eigenvalue which repeats. For a more detailed overview of how the degrees between parts are chosen, see the beginning of Section 5.

$2$-lifts

Given a graph $G$, a signing of $G$ is a map $s:E(G)\to\{\pm 1\}$. Each signing possesses an adjacency matrix $A_{s}$ where entries of $A_{G}$ corresponding to an edge $e$ are replaced with $s(e)$. To any signing $s$ of $G$, one may associate a graph $\tilde{G}$ on the doubled vertex set $\{v_{0},v_{1}\colon v\in V(G)\}$ wherein, for each edge $uv\in E(G)$, edges $u_{0}v_{0}$ and $u_{1}v_{1}$ are drawn in $\tilde{G}$ if $s(uv)=1$, and edges $u_{0}v_{1}$ and $u_{1}v_{0}$ are drawn in $\tilde{G}$ if $s(uv)=-1$. The adjacency matrix of $\tilde{G}$ can be written in block form as

[TABLE]

Such a graph $\tilde{G}$ is called a $2$-lift of $G$, on account of the $2$-covering map $\tilde{G}\to G$ mapping $v_{0}$ and $v_{1}$ to $v$. We make the following observations:

1. (1)

The multiset of the eigenvalues of $A_{\tilde{G}}$ is the disjoint union of that of $A_{G}$ and that of $A_{s}$. If $x$ is an eigenvector of $A_{G}$ (resp. $A_{s}$) with eigenvalue $\mu$, then $x\oplus x$ (resp. $x\oplus-x$) is an eigenvector of $A_{\tilde{G}}$ with the same eigenvalue $\mu$. 2. (2)

If a vertex $v\in V(G)$ has degree $d_{v}$, the vertices $v_{0},v_{1}\in V(\tilde{G})$ above $v$ also have degree $d_{v}$. 3. (3)

If $G$ is $k$-partite, with $V(G)=V_{1}\sqcup V_{2}\sqcup\cdots\sqcup V_{k}$, then $\tilde{G}$ is also $k$-partite; there are no edges between vertices corresponding to elements of $V_{i}$ for each $i$.

Ramanujan lifts

A special case of a theorem of Marcus, Spielman, and Srivastava gives the existence of lifts which, in a very strong sense, add no large eigenvalues.

Theorem 2.1** ([MSS15, Theorem 5.3]).**

Every $d$-regular graph possesses a signing whose adjacency matrix $A_{s}$ has second eigenvalue at most $2\sqrt{d-1}$.

We will make use of the following restatement in the case of bipartite graphs.

Corollary 2.2**.**

For every $d$-regular bipartite graph $G$, there exists a $2$-lift $\tilde{G}$ of $G$ for which, for every eigenvalue $\mu$ of $A_{\tilde{G}}$ satisfying $|\mu|>2\sqrt{d-1}$, the $\mu$-eigenspace of $A_{\tilde{G}}$ has the same dimension as that of $A_{G}$. Moreover, if $\{x_{1},\dots,x_{k}\}$ is a basis of the $\mu$-eigenspace of $A_{G}$, then $\{x_{1}\oplus x_{1},\dots,x_{k}\oplus x_{k}\}$ is a basis of the $\mu$-eigenspace of $A_{G}$.

Proof.

Let $s$ be a signing of $G$ as in Theorem 2.1 so that $\lambda_{1}(A_{s})\leq 2\sqrt{d-1}$. Since $A_{s}$ is a signing of a bipartite graph, its spectrum is symmetric about [math], and so $\|A_{s}\|\leq 2\sqrt{d-1}$. The statement then follows from observation (1) on $2$-lifts. ∎

Multiplicity-incrementing lifts and graph factors

While we need Ramanujan lifts to ensure we don’t have too many large eigenvalues, we will construct more specialized $2$-lifts for the step in which second eigenvalue multiplicity is incremented. Such lifts come from choosing appropriate $a$-factors.

Definition 2.3**.**

Given a $d$-regular graph $G$ and an integer $0\leq a\leq d$, an $a$-factor of $G$ is an $a$-regular subgraph of $G$ on the vertex set $V(G)$. Equivalently, $a$-factors are determined by selection of a subset $S\subset E(G)$ in which every vertex of $G$ is incident to exactly $a$ edges in $S$.

As stated in the outline of our approach (specialized to integer eigenvalue), we shall lift by a signing in which each vertex is incident to the same number of edges signed $-1$. Letting $H$ denote the graph consisting of the edges between two parts which are signed $-1$, this is equivalent to $H$ being a factor of $G$. Consider choosing an $a$-factor $H$ of a $d$-regular graph $G$ at random from a distribution in which each edge appears in $H$ with equal probability, and let $s$ be the signing determined by letting $s(e)=-1$ if and only if $e\in E(H)$. Then

[TABLE]

Given any $H$, $A_{s}$ will have an eigenvector (the all-ones vector) with eigenvalue $d-2a$. On the other hand, if we can choose $H$ so that $A_{s}$ is close in spectral norm to its expectation, then the eigenvalues of $A_{G}$ should “shrink” by a factor of $1-2a/d$ up to some small error, and $d-2a$ should be the largest eigenvalue of $A_{s}$. We are not able to show that $H$ can be chosen in this way. However, in accordance with this goal, we give the following proposition, which gives the existence of a distribution on $a$-factors of a $d$-regular bipartite graph $G$ that, up to a matrix of small spectral norm, displays concentration for Lipschitz functions of the form $u^{\intercal}A_{H}v$. Once we have performed enough Ramanujan lifts, this concentration result will suffices to show the existence of a signing $s$ with $\lambda_{1}(A_{s})=d-2a$.

Proposition 2.4**.**

Let $a$ and $d$ be positive integers with $0\leq a\leq d$. Let $G$ be a bipartite $d$-regular graph on $n$ vertices. There exists a probability distribution on the set of pairs $(H,M)$ where $H$ is an $a$-factor of $G$ and $M$ is an $n\times n$ matrix with $\|M\|\leq 6\sqrt{d}$ such that, for any vectors $u,v\in\mathbb{R}^{n}$ and any real $t\geq 0$,

[TABLE]

The matrix $M$ is more an artifact of the proof than an essential component of the distribution, and one should think of the distribution described in the proposition as a distribution solely on $a$-factors with some “extra data” attached which is useful for analysis.

We conclude this section with an outline of the remainder of the paper. In Section 3, we show Proposition 2.4 and give a corollary which allows us to construct $a$-factors whose adjacency matrices are, on particular subspaces of $\mathbb{R}^{V(G)}$, close to $(a/d)A_{G}$ in spectral norm. In Section 4, we apply this construction to show case (a) of Theorem 1.5, giving large multiplicity of large integer second eigenvalues from regular bipartite graphs. In Section 5, we generalize this construction to $k$-partite graphs and show case (b) of Theorem 1.5. Then, in Section 6, we apply the graphs constructed in Theorem 1.5(b) to the problem of equiangular lines, giving Theorem 1.4.

3. Random factors of graphs

In this section, we show Proposition 2.4, which gives a distribution on $a$-factors $H$ of any $d$-regular bipartite graph $G$ which, up to a matrix of small spectral norm, displays relatively tight concentration. The distribution is constructed as follows:

1. (1)

In the special case when $d$ is even and $a=d/2$:

1. (a)

Partition $G$ into edge-disjoint cycles, at each step adding the smallest remaining cycle to the partition. Lemma 3.1 shows that the union of the large cycles has small spectral norm. 2. (b)

Note that all cycles in the partition have even length. From each cycle with $2\ell$ edges, choose an alternating set of $\ell$ edges uniformly at random from the two possible choices and add this set to $H$. For large cycles, one may make the choices deterministically. For example, if one such cycle is $v_{1}v_{2}v_{3}v_{4}v_{5}v_{6}$, add either the edges $\{v_{1}v_{2},v_{3}v_{4},v_{5}v_{6}\}$ or the edges $\{v_{2}v_{3},v_{4}v_{5},v_{6}v_{1}\}$ to $H$. 2. (2)

Using (1) as a subroutine, proceed recursively:

1. (a)

In the base case of $d=1$, there is nothing to choose: $G$ has a unique [math]-factor and a unique $1$-factor. 2. (b)

If $a>d/2$, construct a random $(d-a)$-factor and take its complement in $G$. 3. (c)

If $a\leq d/2$ and $d$ is even, construct a random $(d/2)$-factor $G^{\prime}$ of $G$ using (1), and then construct a random $a$-factor of the $d/2$-regular bipartite graph $G^{\prime}$. 4. (d)

If $d$ is odd, remove a $1$-factor (i.e. a perfect matching) from $G$ arbitrarily to form a $(d-1)$-regular graph $G^{\prime}$, and then find a random $a$-factor of $G^{\prime}$.

The terms arising from the large cycles in step 1(b) and from the arbitrary perfect matchings in 2(d) are absorbed into the auxiliary matrix $M$. What is left is a sum of many “small” random choices, and so concentration arises from Chernoff-style arguments.

To prove our concentration result, we will first require the following technical lemma, which assists in bounding the spectral radius of $M$.

Lemma 3.1**.**

Let $G$ be a graph on $n$ vertices with maximum degree at most $d$ and no cycles of length at most $2\lceil\log_{2}n\rceil$. Then $\|A_{G}\|\leq(2\sqrt{2})\sqrt{d}$.

Proof.

Let $r=\lceil\log_{2}n\rceil$. On one hand, we have

[TABLE]

On the other hand, $\operatorname{tr}A_{G}^{2r}$ counts the number of closed walks of length $2r$ in $G$. Each such walk is also a walk on the universal cover of $G$, since otherwise would imply the existence of a cycle of $G$ of length at most $2r$. Since every vertex has maximum degree $d$, the number of such walks starting at a given vertex is at most $d^{r}C_{r}\leq(4d)^{r}$, where $C_{r}$ is a Catalan number. This gives

[TABLE]

and so

[TABLE]

We now state and prove the concentration result for step (1) of the above outline.

Lemma 3.2**.**

Let $d$ be an even positive integer. Let $G$ be a bipartite $d$-regular graph on $n$ vertices. There exists a (deterministic) $n\times n$ matrix $M$ satisfying $\|M\|\leq\sqrt{2d}$ and a probability distribution on the set of $d/2$-factors $H$ of $G$ such that, for any vectors $u,v\in\mathbb{R}^{n}$ and any real $t\geq 0$,

[TABLE]

Proof.

First, partition the edges $E=E(G)$ of $G$ into sets $E_{0}$ and $E_{1}$ as follows: while there is a cycle in $G$ of length at most $2\lceil\log_{2}n\rceil$, add the edges of this cycle to $E_{1}$ and remove them from $G$; once no such cycle exists, add all remaining edges to $E_{0}$. Let $G_{0}$ and $G_{1}$ be the graphs on the same vertex set as $G$ with edge sets $E_{0}$ and $E_{1}$, respectively. By Lemma 3.1, $\|A_{G_{0}}\|\leq 2\sqrt{2d}$.

We construct $H$ by partitioning $E_{0}$ deterministically and $E_{1}$ randomly. Since $E_{1}$ is a union of cycles, every vertex has even degree in $G_{1}$ and thus in $G_{0}$, and so the edges of $E_{0}$ can be partitioned into cycles. For each such cycle, pick an arbitrary alternating set of edges of the cycle as in step 1(b) above; let the resulting edges be $E_{0}^{\prime}$ and let $H_{0}$ be the graph with edge set $E_{0}^{\prime}$. Each vertex has degree in $H_{0}$ half of that in $G_{0}$. Let $M=1/2\cdot A_{G_{0}}-A_{H_{0}}$; since $M$ is half the adjacency matrix of a signing of $G_{0}$, $\|M\|\leq(1/2)\|A_{G_{0}}\|\leq\sqrt{2d}$.

Now, $G_{1}$ is a union of cycles of length at most $2\lceil\log_{2}n\rceil$. For each such cycle, choose an alternating set of half the edges randomly from the two possible choices, and add its edges to $E_{1}^{\prime}$; let $H_{1}$ be the graph with edge set $E_{1}^{\prime}$. Let $H$ be the graph with edge set $E_{0}^{\prime}\cup E_{1}^{\prime}$; $H$ is an $a$-factor of $G$, and

[TABLE]

Consider the random variable

[TABLE]

Letting $\mathcal{C}$ be the set of cycles the union of which is $E_{1}$,

[TABLE]

where $X_{C}\sim\operatorname{Unif}(\{\pm 1\})$ and $\tilde{A}_{C}$ is the adjacency matrix of a signing of $C$ in which incident edges are signed alternately. Since $u^{\intercal}\tilde{A}_{C}v$ is a sum of $2\operatorname{len}(C)$ terms of the form $\pm u_{i}v_{j}$ for integers $i$ and $j$,

[TABLE]

So, the bounded differences inequality implies, for any $\mu\geq 0$,

[TABLE]

We have

[TABLE]

so

[TABLE]

This implies the desired concentration upon setting $\mu=t\sqrt{dn\lceil\log_{2}n\rceil}\|u\|_{\infty}\|v\|_{\infty}$. ∎

We use this lemma as a black box, as described in step (2) of the outline, to show Proposition 2.4.

Proof of Proposition 2.4.

Let $c_{d}$ be the sequence defined recursively by $c_{1}=0$, $c_{d}=c_{d-1}+1$ if $d>1$ is odd, and $c_{d}=c_{d/2}+\sqrt{2d}$ if $d$ is even. We proceed by strong induction on $d$ to show that there exists a distribution on pairs $(H,M)$ of $a$-factors of $G$ and $n\times n$ matrices wherein $\|M\|\leq c_{d}$ and, for any $t\geq 0$ and $u,v\in\mathbb{R}^{n}$,

[TABLE]

In the base case of $d=1$, there is not much choice; if $a=0$ then $H$ is empty, while if $a=1$ then $H=G$. In either case, we can take $M=0$, and $u^{\intercal}(A_{H}-a/d\cdot A_{G}+M)v=0$ always.

For the inductive step, we first treat the case where $d$ is even and $a\leq d/2$. In this case, we proceed in two steps:

1. (1)

Construct a $d/2$-factor $G_{1}$ of $G$ via Lemma 3.2, and let $M_{1}$ be the associated matrix. 2. (2)

Construct an $a$-factor $H$ of $G_{1}$ via the inductive hypothesis, and let $M_{2}$ be the associated matrix.

We set $M=M_{2}+2a/d\cdot M_{1}$. We have $\|M_{1}\|\leq\sqrt{2d}$ by Lemma 3.2 and $\|M_{2}\|\leq c_{d/2}$ by the inductive hypothesis. This gives

[TABLE]

and

[TABLE]

as well as the concentration inequalities

[TABLE]

This implies by the union bound that the probability that $u^{\intercal}(A_{H}-a/d\cdot A_{G}+M)v$ exceeds

[TABLE]

in magnitude is at most $2(\log_{2}d)e^{-t^{2}/2}$, as this would imply that one of the two above events occurs. Since

[TABLE]

this is sufficient.

In the case where $d$ is even and $a>d/2$, we first construct a pair $(H^{\prime},M^{\prime})$ where $H^{\prime}$ is a $(d-a)$-factor of $G$. Let $H$ be the complement of $H^{\prime}$ in $G$, and let $M=-M^{\prime}$. Since

[TABLE]

the hypothesis for $(d-a)$-factors of $d$-regular graphs suffices.

Finally, we treat the case where $d$ is odd. If $a>d/2$ then we can proceed as in the previous paragraph from the $a<d/2$ case, so assume $a<d/2$. We proceed as follows:

1. (1)

Select an arbitrary perfect matching $H^{\prime}\subset G$, and let $G_{1}$ be the complement of $H^{\prime}$ in $G$, a $(d-1)$-factor of $G$. 2. (2)

Construct an $a$-factor $H$ of $G_{1}$ via the inductive hypothesis, and let $M_{1}$ be the associated matrix.

Define $M=\frac{a}{d}A_{H^{\prime}}-\frac{a}{d(d-1)}A_{G_{1}}+M_{1}$. We have

[TABLE]

as well as

[TABLE]

so the inductive hypothesis at $d-1$ suffices.

All that remains is to show that $c_{d}\leq 6\sqrt{d}$ for all $d$. We can show

[TABLE]

by induction on $d$, so the result follows from

[TABLE]

which is easy to check. ∎

We conclude this section with a corollary of Proposition 2.4 which shows a concentration result for factors of regular bipartite graphs on subspaces with an orthonormal basis with small $L^{\infty}$ norm. This is what we will use when constructing our multiplicity-incrementing lifts.

Corollary 3.3**.**

Let $a$ and $d$ be positive integers with $0\leq a\leq d$. Let $G$ be a bipartite $d$-regular graph on $n$ vertices, and let $1\leq K,m\leq n$. Let $S$ be a finite set containing $V(G)$, and let $U\subset\mathbb{R}^{S}$ be a subspace of dimension at most $m$ spanned by pairwise orthogonal unit vectors $u_{1},u_{2},\dots,u_{m^{\prime}}$ satisfying $\|u_{i}\|_{\infty}\leq\sqrt{K/n}$ for each $1\leq i\leq m^{\prime}$. If $m\geq\max(2\log_{2}d,15)$ and $n\geq K^{3}m^{6}$, then there exists an $a$-regular subgraph $H\subset G$ on $V(G)$ satisfying

[TABLE]

for every unit vector $v\in U$, where $A_{H}$ and $A_{G}$ are extended by zeros to a matrix in $\mathbb{R}^{S\times S}$.

Proof.

For each $1\leq i\leq m^{\prime}$, let $\overline{u_{i}}$ be the restriction of $u_{i}$ to $\mathbb{R}^{V(G)}$; note that $\|\overline{u_{i}}\|_{\infty}\leq\|u_{i}\|_{\infty}$. Set $t=\sqrt{6.5\ln m}$, so that

[TABLE]

By Proposition 2.4 and the union bound, there exists an $a$-regular subgraph $H\subset G$ on $V(G)$ and an $n\times n$ matrix $M$ satisfying $\|M\|\leq 4\sqrt{d}$ for which

[TABLE]

for all $1\leq i,j\leq m^{\prime}$. Now, take any unit vector $v\in U$. Since $\{u_{i}\}_{i}$ is an orthonormal basis of $U$, we can write $v=\sum_{i=1}^{m^{\prime}}c_{i}u_{i}$ for some real $c_{1},\dots,c_{m^{\prime}}$ with $\sum c_{i}^{2}=1$. This gives

[TABLE]

where we have used the Cauchy–Schwarz inequality. Since $n\geq 2$, $n/\lceil\log_{2}n\rceil\geq n^{2/3}$, and since $m\geq 15$, $9\sqrt{\ln m}\leq m$, so this bound gives

[TABLE]

by our bound on $n$. The fact that $\|M\|\leq 6\sqrt{d}$ finishes the proof. ∎

4. Integer second eigenvalue

In this section, we use Corollary 3.3 to construct fixed-degree regular graphs with a fixed second eigenvalue of multiplicity $\Omega(\log\log n)$; the fixed eigenvalue can be any sufficiently large integer. The main result of this section will be the following “incrementing” proposition. Iterating this will allow us to obtain Theorem 1.5(a).

Proposition 4.1**.**

Let $d\geq 22000$, and let $a=\lceil 12\sqrt{d}\rceil$. Suppose $G$ is a bipartite $d$-regular graph on $n$ vertices for which $\lambda_{2}(A_{G})\leq d-2a$, and let $\ell$ be the multiplicity of $d-2a$ as an eigenvalue of $A_{G}$. Then there exists a bipartite $d$-regular graph $\tilde{G}$ on at most $4n^{9}$ vertices with second eigenvalue $d-2a$ of multiplicity at least $\ell+1$.

The process will be, as described at the beginning of Section 2, to first perform many lifts which add no new large eigenvalue, and then to apply a multiplicity-incrementing lift determined by a random $a$-factor of $G$ using Corollary 3.3. The properties of this lift are described in the following lemma.

Lemma 4.2**.**

Let $d\geq 22000$, and let $a=\lceil 12\sqrt{d}\rceil$. Let $G$ be a $d$-regular bipartite graph with $n$ vertices satisfying $\lambda_{2}(A_{G})\leq d-2a$. Suppose that there are at most $m$ eigenvalues of $G$ greater than $2\sqrt{d-1}$, and $A_{G}$ possesses an orthonormal eigenbasis in which each component of these $m^{\prime}\leq m$ top eigenvectors is at most $\sqrt{m/n}$ in magnitude. Then, as long as $m\geq 2\log_{2}d$ and $n>m^{9}$, there exists an $a$-factor $H$ of $G$ with $\lambda_{1}(A_{G}-2A_{H})=d-2a$.

We first prove the proposition given this lemma, and then prove the lemma.

Proof of Proposition 4.1.

Let $t$ be a positive integer. By applying Corollary 2.2 $t$ times, we can find a $d$-regular bipartite graph $G^{\prime}$ on $n^{\prime}:=n2^{t}$ vertices for which, for each eigenvalue $\mu$ of $A_{G^{\prime}}$ with $|\mu|\geq 2\sqrt{d-1}$, the $\mu$-eigenspace of $A_{G^{\prime}}$ is exactly

[TABLE]

In particular,

[TABLE]

and $A_{G^{\prime}}$ has eigenvalue $d-2a$ with multiplicity $\ell$. Moreover, $A_{G}^{\prime}$ possesses at most $|V(G)|=n$ eigenvalues greater than $2\sqrt{d-1}$, and an orthonormal eigenbasis of $A_{G^{\prime}}$ containing

[TABLE]

possesses no eigenvector with an eigenvalue larger than $2\sqrt{d-1}$ and an entry larger than $2^{-t/2}=\sqrt{n/n^{\prime}}$ in magnitude. So, as long as $n\geq 2\log_{2}d$ (which holds since there exists a $d$-regular graph $G$ on $n$ vertices) and $n2^{t}>n^{9}$, we can apply Lemma 4.2 to find an $a$-factor $H$ of $G^{\prime}$ with $\lambda_{1}(A_{G^{\prime}}-2A_{H})=d-2a$. Letting $s$ be the associated signing of $G^{\prime}$ and $\tilde{G}$ be the associated $2$-lift, the spectrum of $\tilde{G}$ contains the eigenvalue $d-2a$ with multiplicity at least $\ell+1$. Selecting $t$ to be the smallest positive integer satisfying $2^{t}>n^{8}$ finishes the proof. ∎

Proof of Lemma 4.2.

Let $U$ be the span of all eigenvectors of $A_{G}$ with eigenvalue exceeding $2\sqrt{d-1}$ and strictly less than $d$; note that $\dim U\leq m^{\prime}\leq m$, and $U$ possesses an orthonormal basis in which each vector has $L^{\infty}$ norm at most $\sqrt{m/n}$. So, by Corollary 3.3 (which we can apply since $2\log_{2}d>15$), there exists an $a$-factor $H$ of $G$ with

[TABLE]

for every unit vector $v\in U$. In particular,

[TABLE]

Now, since $H$ is $a$-regular, the matrix $A_{G}-2A_{H}$ has eigenvalue $d-2a$ with the all-ones vector as an eigenvalue. We need to show that there are no larger eigenvalues. To this end, consider any vector $w$ orthogonal to the all-ones vector; it suffices to show $w^{\intercal}A_{H}w\leq d-2a$. Write $w=x+y$, where $x\in U$ and $y\in U^{\prime}$, the span of all eigenvectors of $A_{G}$ with eigenvalue at most $2\sqrt{d-1}$. We compute

[TABLE]

where we have used ($\star$ ‣ 4) and the fact that $\|A_{H}\|=a$. Now, this allows us to bound

[TABLE]

This is a quadratic form in $\|x\|$ and $\|y\|$. The maximum of a quadratic form $Rx^{2}+2Sxy+Ty^{2}$ over the unit circle can be easily computed to be $(R+T+\sqrt{(R-T)^{2}+4S^{2}})/2$. Since $\|x\|^{2}+\|y\|^{2}=\|w\|^{2}=1$, all that remains is to show that this is at most $d-2a$ for the quadratic form above in $\|x\|$ and $\|y\|$ as long as $d\geq 22000$. This is a simple computation. ∎

We conclude this section by establishing Theorem 1.5(a).

Proof of Theorem 1.5(a).

Let $\ell\geq 20000$ be a positive integer. We will show that, for infinitely many $n$, there exist bounded-degree regular graphs on $n$ vertices with second eigenvalue exactly $\ell$, of multiplicity $\Omega(\log\log n)$.

Let $d$ be a positive integer exceeding $22000$ for which $d-2\lceil 12\sqrt{d}\rceil=\ell$ and let $a=\lceil 12\sqrt{d}\rceil$, so that $\ell=d-2a$. Set $G_{0}=K_{d,d}$ and $n_{0}=2d$, and define a sequence $(n_{i})$ recursively by $n_{i+1}=4n_{i}^{9}$. Note that $\lambda_{2}(G_{0})=0<d-2a$. Repeated applications of Proposition 4.1 show that, for each $i\geq 1$, there exists a bipartite $d$-regular graph $G_{i}$ such that

1. (i)

$G_{i}$ has at most $n_{i}$ vertices, and 2. (ii)

$d-2a$ is the second eigenvalue of $G_{i}$ and has multiplicity at least $i$.

The fact that $n_{i}=2^{2^{\Theta(i)}}$ finishes the proof, since $i=\Theta(\log\log n_{i})$. ∎

Remark 4.3**.**

If we were able to find, for any bipartite $d$-regular graph $G$, an $a$-factor $H$ with

[TABLE]

this would allow us to remove the Ramanujan lift step and attain second eigenvalue multiplicity on the order of $\log n$. This aim is reminiscent of a result of Bilu and Linial [BL06, Theorem 3.1] that there exists a signing $s$ of such a graph with $\|A_{s}\|=O(d^{1/2}\log^{3/2}d)$, but in our distribution there seems not to be enough randomness to follow their spectral radius-bounding framework.

5. Non-integer second eigenvalue

In this section we give a generalization of the framework from the previous section, which we will use to prove our main result. We begin by elaborating on the sketch of the $k$-partite graphs-based strategy given in Section 2. To do this, we first need some terminology.

Definition 5.1**.**

Recall that a $k\times k$ symmetric matrix $M$ is irreducible if the (not necessarily simple) graph with adjacency matrix $M$ is connected. (Irreducibility ensures that we can apply the Perron–Frobenius theorem to $M$.) Given a symmetric $k\times k$ irreducible matrix $M$ with zeros on the diagonal and nonnegative integer entries:

• •

A graph lift of $M$ is a lift of the non-simple graph with adjacency matrix $M$. In other words, it is a $k$-partite graph $G$ on vertex set $V_{1}\sqcup V_{2}\sqcup\cdots\sqcup V_{k}$ such that, for each $1\leq i,j\leq k$, $G[V_{i}\sqcup V_{j}]$ is a $M_{ij}$-regular bipartite graph.

• •

A sign matrix of $M$ is a symmetric integer matrix $M^{\prime}$ for which $M^{\prime}_{ij}\equiv M_{ij}\pmod{2}$ and $|M^{\prime}_{ij}|\leq M_{ij}$ for each $1\leq i,j\leq k$.

• •

An $M^{\prime}$-signing of a graph lift $G$ of $M$ is an assignment of an element of $\{\pm 1\}$ to each edge of $G$ such that, for each $1\leq i,j\leq k$, the labels of the edges of $G[V_{i}\sqcup V_{j}]$ incident to each vertex of $V_{i}\sqcup V_{j}$ sum to $M^{\prime}_{ij}$.

For example, take

[TABLE]

for $a=\lceil 12\sqrt{d}\rceil$. Then $M^{\prime}$ is a sign matrix of $M$, and the graph lifts of $M$ are exactly the $d$-regular bipartite graphs. Lemma 4.2 demonstrates the existence of an $M^{\prime}$-signing of such a graph with top eigenvalue exactly $d-2a$, under certain conditions. We now prove the following lemma, which gives information about the eigendata of graph lifts and matrix-signings.

Lemma 5.2**.**

Let $(M,M^{\prime},G)$ be as in Definition 5.1.

1. (a)

The top eigenvalue of $A_{G}$ is $\lambda_{1}(M)$; if $v$ is a top eigenvector of $M$, then the vector which assigns each vertex in $V_{i}$ the value $v_{i}$ is an eigenvector of $A_{G}$ with this eigenvalue. 2. (b)

Consider an $M^{\prime}$-signing $s:E(G)\to\{\pm 1\}$. The eigenvalues of $M^{\prime}$ are eigenvalues of the adjacency matrix $A_{s}$ of this signing, and moreover their eigenspaces span the space of vectors $v$ in $\mathbb{R}^{V(G)}$ which are constant on each part of the $k$-partition.

Proof.

For (a), it is easy to check that $\lambda_{1}(M)$ is an eigenvalue of $A_{G}$ with the eigenvector described as such. The fact that this is the top eigenvalue follows from the observation that, since both $A_{G}$ and $M$ possess nonnegative entries, $v$ is a Perron eigenvector, and so $\lambda_{1}(M)$ is the Perron eigenvalue of $A_{G}$. For (b), it is easy to check that if $w$ is an eigenvector of $M^{\prime}$ with eigenvalue $\mu$, the vector which assigns a vertex in $V_{i}$ the value $w_{i}$ is an eigenvector of $A_{s}$ with eigenvalue $\mu$ as well. ∎

Due to this lemma, the eigenvalue for which we are able to guarantee large multiplicity is $\lambda_{1}(M^{\prime})$. We know that it will appear in any $M^{\prime}$-signing; all that remains is to show that, under certain conditions, no eigenvalue is larger. We first make these conditions explicit.

Definition 5.3**.**

Denote by $\mathcal{M}$ the set of triples $(M,M^{\prime},\beta)$ for which

1. (a)

$M$ is an irreducible $k\times k$ nonnegative integer symmetric matrix with zeros on the diagonal, and $M^{\prime}$ is a sign matrix of $M$, 2. (b)

$\beta\in(0,1/2)$, and 3. (c)

if $D=(M-M^{\prime})/2$, $\gamma=\lambda_{1}(|D-\beta M|)$ (where the absolute value is taken entrywise), and $\sigma=\sum_{i,j}\sqrt{M_{ij}}$, then

[TABLE]

We may now state the following proposition which generalizes Proposition 4.1.

Proposition 5.4**.**

Suppose $(M,M^{\prime},\beta)\in\mathcal{M}$. Let $G$ be a graph lift of $M$ on $n$ vertices for which $\lambda_{2}(A_{G})\leq\lambda_{1}(M^{\prime})$, and let $\ell$ be the multiplicity of $\lambda_{1}(M^{\prime})$ as an eigenvalue of $A_{G}$. Then there exists a graph lift $\tilde{G}$ of $G$ on at most $4n^{9}$ vertices with second eigenvalue $\lambda_{1}(M^{\prime})$ of multiplicity at least $\ell+1$.

As in the proof of Proposition 4.1, the main technical piece will be a lemma which gives the existence of a multiplicity-incrementing lift. We apply such a lift after applying enough Ramanujan lifts between each pair of parts.

Lemma 5.5**.**

Suppose $(M,M^{\prime},\beta)\in\mathcal{M}$, and let $G$ be a graph lift of $M$ on $n$ vertices. Let $m$ be a positive integer. Define $\sigma=\sum_{i,j}\sqrt{M_{ij}}$, as in Definition 5.3. Suppose

1. (a)

$\lambda_{2}(A_{G})\leq\lambda_{1}(M^{\prime})$, 2. (b)

at most $m$ eigenvalues of $G$ exceed $\sigma$, 3. (c)

$A_{G}$* possesses an orthonormal eigenbasis, a subset of which is a basis for those vectors constant on each part of the $k$-partition of $V(G)$ afforded by $M$, in which each component of each eigenvector with eigenvalue greater than $\sigma$ is at most $\sqrt{m/n}$ in magnitude, and* 4. (d)

$n>m^{9}$.

Then there exists a $M^{\prime}$-signing $s:E(G)\to\{\pm 1\}$ with adjacency matrix $A_{s}$ satisfying $\lambda_{1}(A_{s})=\lambda_{1}(M^{\prime})$.

Each of the conditions in this lemma parallels a condition in Lemma 4.2. In particular, the condition that $(M,M^{\prime},\beta)\in\mathcal{M}$ analogizes the degree bound $d\geq 22000$ and the selection of $a=\lceil 12\sqrt{d}\rceil$. The proof will be quite similar to that of Lemma 4.2.

Proof.

As in Definition 5.3, define $D=(M-M^{\prime})/2$ and $\gamma=\lambda_{1}(|D-\beta M|)$. Let $V_{1}\sqcup V_{2}\sqcup\cdots\sqcup V_{k}$ be the $k$-partition of $V(G)$ afforded by $M$. Let $U_{0}\subset\mathbb{R}^{V(G)}$ be the space of vectors which are constant on each $V_{i}$. By Lemma 5.2, the top eigenvalue of $G$ is $\lambda_{1}(M)$, and its eigenvector lies in $U_{0}$. Also by Lemma 5.2, we know that for any $M^{\prime}$-signing $s$ of $G$ there exists a basis of $U_{0}$ which is contained within an eigenbasis of $A_{s}$, one vector in this basis contributes an eigenvalue $\lambda_{1}(M^{\prime})$ to $A_{s}$, and no vector in this basis contributes a larger eigenvalue. So, it suffices to find an $M^{\prime}$-signing $s$ for which, for any vector $w\in\mathbb{R}^{V(G)}$ orthogonal to $U_{0}$, $w^{\intercal}A_{s}w\leq\lambda_{1}(M^{\prime})$.

We construct this signing separately on the edges between $V_{i}$ and $V_{j}$ for each pair $(i,j)$. Let $G_{ij}$ be the graph $G[V_{i}\sqcup V_{j}]$ on vertex set $V(G)$, so that

[TABLE]

Let $v_{2},\dots,v_{m^{\prime}}$ be the eigenvectors of $A_{G}$ in the partition described in c with eigenvalues exceeding $\sigma$ and which do not lie in $U_{0}$, and let $U_{1}$ be the subspace of $\mathbb{R}^{V(G)}$ they span.

By b, $\dim U_{1}\leq m^{\prime}<m$. The space $U_{1}$ satisfies the condition of Corollary 3.3 with $K=m$. So, since $n\geq m^{9}$ (condition d), we can for each $1\leq i<j\leq k$ find a $D_{ij}$-factor $H_{ij}$ of $G_{ij}$ which satisfies

[TABLE]

for every $v\in U_{1}$. We conclude, letting $H$ be the graph formed by the union of the edge sets $H_{ij}$ and letting $B=\sum\frac{D_{ij}}{M_{ij}}A_{G_{ij}}$, that

[TABLE]

for every $v\in U_{1}$. Let $s$ be the signing for which edges in $H_{ij}$ for any $(i,j)$ are assigned $-1$ and all other edges are assigned $1$. By definition, $s$ is an $M^{\prime}$-signing; we also have $A_{s}=A_{G}-2A_{H}$.

Now, let $w\in\mathbb{R}^{V(G)}$ be any unit vector orthogonal to $U_{0}$; write $w=x+y$, where $x\in U_{1}$ and $y$ lies in the orthogonal complement of $U_{0}+U_{1}$; note that $\|x\|^{2}+\|y\|^{2}=1$. We first see

[TABLE]

Since $H$ is a graph lift of $D$, and the entries of $D$ are nonnegative integers, Lemma 5.2(a) implies that $\|A_{H}\|=\lambda_{1}(A_{H})=\lambda_{1}(D)$. We now upper-bound $\|B-\beta A_{G}\|$. For any unit vector $u\in\mathbb{R}^{V(G)}$, write $u=u_{1}+\cdots+u_{k}$, where $u_{i}$ is supported on $V_{i}$. We have, similarly to the proof of Lemma 5.2,

[TABLE]

since $\|u_{1}\|^{2}+\cdots+\|u_{k}\|^{2}=1$. So, $\|B-\beta A_{G}\|\leq\gamma$. This implies that

[TABLE]

where we have used condition a in the last bound, since $x\in U_{0}$. Since we have for any $R,S,T$ and any $\|x\|,\|y\|\in\mathbb{R}$ with $\|x\|^{2}+\|y\|^{2}=1$,

[TABLE]

the result follows from the definition of $\mathcal{M}$. ∎

Proof of Proposition 5.4.

Armed with the above lemma, the proof will be very similar to that of Proposition 4.1. Let $t$ be a nonnegative integer. We will first construct a graph lift $G_{t}$ of $M$ on $n_{t}:=n2^{t}$ vertices for which, for each eigenvalue $\mu$ of $A_{G_{t}}$ with $\mu\geq\sigma$, the $\mu$-eigenspace of $A_{G_{t}}$ is exactly

[TABLE]

Indeed, starting with $G_{0}=G$ when $t=0$, we can construct these graphs by successively applying Ramanujan $2$-lifts between each pair of parts. Let $V_{1}\sqcup V_{2}\sqcup\cdots\sqcup V_{k}$ be the $k$-partition of $V(G_{t})$ afforded by $M$. For each $1\leq i<j\leq k$, let $s_{ij}:E(G_{t}[V_{i}\sqcup V_{j}])\to\{\pm 1\}$ be a signing of the $M_{ij}$-regular bipartite graph $G_{t}[V_{i}\sqcup V_{j}]$ whose adjacency matrix has second eigenvalue at most $2\sqrt{M_{ij}-1}<2\sqrt{M_{ij}}$, guaranteed to exist by Theorem 2.1 (if $M_{ij}=0$, then $s_{ij}$ is trivial and its adjacency matrix has second eigenvalue $0=2\sqrt{M_{ij}}$). Construct a signing $s:E(G_{t})\to\{\pm 1\}$ by combining the partial signings $s_{ij}$, so that $A_{s}=\sum_{1\leq i<j\leq k}A_{s_{ij}}$ has top eigenvalue at most

[TABLE]

Then the $2$-lift $G_{t+1}$ of $G_{t}$ associated to $s$ has no eigenvalues exceeding $\sigma$ which do not come from $A_{G_{t}}$, and so $G_{t+1}$ satisfies the desired properties. In particular, for each $t$,

[TABLE]

(we know $\sigma\leq\lambda_{1}(M^{\prime})$ by condition (c) in the definition of $\mathcal{M}$), and $A_{G_{t}}$ has eigenvalue $\lambda_{1}(M^{\prime})$ with multiplicity $k$. Identically to the proof of Proposition 4.1, conditions (b) and (c) of Lemma 5.5 are satisfied by $G_{t}$, since we have described the eigenspaces of $A_{G_{t}}$ with large eigenvalue. So, we can apply this lemma to get an $M^{\prime}$-signing $s:E(G)\to\{\pm 1\}$ with $\lambda_{1}(A_{s})=\lambda_{1}(M^{\prime})$ as long as $2^{j}>n^{8}$. The spectrum of the associated $2$-lift contains the eigenvalue $\lambda_{1}(M^{\prime})$ with multiplicity at least $\ell+1$, and so selecting $t$ to be the smallest positive integer satisfying $2^{t}>n^{8}$ finishes the proof. ∎

These multiplicity-incrementing steps give us graphs with large multiplicity of fixed second eigenvalue.

Corollary 5.6**.**

Suppose $(M,M^{\prime},\beta)\in\mathcal{M}$ and that there exists a connected graph lift $G_{0}$ of $M$ on at most $r$ vertices with $\lambda_{2}(A_{G_{0}})\leq\lambda_{1}(M^{\prime})$. Then there exists an infinite sequence $r=n_{0}<n_{1}<\cdots$ of positive integers, each satisfying $n_{i+1}\leq 4n_{i}^{9}$, for which there exists a graph on $n_{i}$ vertices with top eigenvalue $\lambda_{1}(M)$ and second eigenvalue $\lambda_{1}(M^{\prime})$ with multiplicity at least $i=\Omega(\log\log n_{i})$. Furthermore, if the all-ones vector is an eigenvector of $M$, then these graphs can be chosen to be regular.

Proof.

This parallels the proof of Theorem 1.5(a). In the previous section. Let $n_{0}=r$, and define a sequence $(n_{i})$ recursively so that $n_{i+1}$ is the unique integer in the interval $(2n_{i}^{9},4n_{i}^{9}]$ for which $n_{i+1}/n_{i}$ is a power of $2$. Repeated applications of Proposition 5.4 show that, for each $i\geq 1$, there exists a graph lift $G_{i}$ of $M$ such that

1. (i)

$G_{i}$ has $n_{i}$ vertices, and 2. (ii)

$\lambda_{1}(M^{\prime})$ is the second eigenvalue of $G_{i}$ and has multiplicity at least $i$.

Since $G_{i}$ is a graph lift of $M$, its top eigenvalue is $\lambda_{1}(M)$ by Lemma 5.2(a). If the all-ones vector is an eigenvector of $M$, then it is the Perron eigenvector of $A_{G_{i}}$ (again by Lemma 5.2(a)) for each $i$, and so (since each $G_{i}$ will be connected) each $G_{i}$ is regular with degree $\lambda_{1}(M)$. ∎

To conclude the proof of Theorem 1.5(b), we need to choose our matrices $M$ and $M^{\prime}$ and show that they satisfy the necessary properties. We do this in the following lemma.

Lemma 5.7**.**

Let $t>u$ be positive integers of the same parity. Define the matrices

[TABLE]

and let $\beta=(t-u)/(2t)$. Suppose $(5/6)t\leq u\leq t-56\sqrt{t}-62$. Then

1. (a)

$\lambda_{1}(M^{\prime})=2\sqrt{u^{2}+1}-1$, 2. (b)

there exists a (connected) graph lift $G_{0}$ of $M$ on $4t$ vertices with $\lambda_{2}(A_{G_{0}})\leq\lambda_{1}(M^{\prime})$, and 3. (c)

$(M,M^{\prime},\beta)\in\mathcal{M}$.

Proof.

We first note that the bounds on $u$ imply that $\beta\leq 1/12$ and $t,u\geq 10^{4}$. To prove part (a), it suffices to compute the eigenvalues of $M^{\prime}$ explicitly to be

[TABLE]

Since $u\geq 2$, the largest of these is $2\sqrt{u^{2}+1}-1$.

To prove part (b), we construct such a graph. Between the parts to be connected by $t$-regular bipartite graphs, we place complete bipartite graphs, forming a graph $G_{0}^{(1)}$. Then we place $3$-regular bipartite graphs between the other two pairs of parts arbitrarily (this is possible since $t\geq 3$); let the union of these two $3$-regular bipartite graphs be $G_{0}^{(2)}$. Since $G_{0}^{(2)}$ is $3$-regular, its adjacency matrix has spectral norm at most $3$. Weyl’s inequality thus tells us

[TABLE]

The spectrum of $G_{0}^{(1)}$ consists of one copy each of $\pm 2t$ and $4t-2$ copies of [math]. This shows $\lambda_{2}(A_{G_{0}})\leq 3\leq\lambda_{1}(M^{\prime})$, as desired.

Finally, we show (c). It is clear that $M^{\prime}$ is a sign matrix of $M$; the rest requires some computation. We have the matrices

[TABLE]

this gives $\lambda_{1}(D)\leq 2t\beta+3$ and $\gamma=\lambda_{1}(|D-\beta M|)\leq 3$. Also, $\sigma=8\sqrt{t}+4\sqrt{3}<8\sqrt{t}+7$. Since the top eigenvalue of positive matrices may only increase if entries are increased, it suffices to show, using that $2u-1\leq\lambda_{1}(M^{\prime})\leq 2u$,

[TABLE]

By sub-additivity of top eigenvalue, we have

[TABLE]

where we have used the computation that

[TABLE]

for $\beta\leq 1/12$. We thus need

[TABLE]

which holds by assumption. ∎

We conclude with the proof of Theorem 1.5(b). In fact, we following prove the (ever so slightly) stronger statement; we will need its precision to show Theorem 1.4.

Theorem 5.8**.**

Let $u\geq 10^{5}$ be an integer. There exists some integer $d\leq 5u/2$ for which, for an infinite sequence $n_{0}<n_{1}<\cdots$ of positive integers, each satisfying $n_{i+1}\leq 4n_{i}^{9}$, there exists a $d$-regular graph on $n_{i}$ vertices with second eigenvalue exactly $2\sqrt{u^{2}+1}-1$ of multiplicity $i=\Omega(\log\log n_{i})$.

Proof.

Let $t$ be a positive integer of the same parity as $u$ satisfying $5t/6\leq u\leq t-56\sqrt{t}-62$, which exists since $u\geq 10^{5}$. Define

[TABLE]

and $\beta=(t-u)/(2t)$. By Lemma 5.7, $\lambda_{1}(M^{\prime})=2\sqrt{u^{2}+1}-1$, there exists a connected graph lift $G_{0}$ of $M$ on $4t$ vertices with $\lambda_{2}(A_{G_{0}})\leq\lambda_{1}(M^{\prime})$, and $(M,M^{\prime},\beta)\in\mathcal{M}$. The fact that the all-ones vector is an eigenvector of $M$ with eigenvalue $2t+3\leq 12u/5+3\leq 5u/2$ is enough to finish the proof by appealing to Corollary 5.6. ∎

6. Application: Equiangular lines

We apply Theorem 5.8 to the problem of equiangular lines with fixed angle. Recall that, if the spectral radius order $k(\lambda)$ of $\lambda=(1-\alpha)/(2\alpha)$ is finite, then $N_{\alpha}(n)\geq(1+\varepsilon)n$ for some $\varepsilon>0$ and large $n$. So, to prove Theorem 1.4, we need to find $\lambda$ which is not the top eigenvalue of any graph, but which may still be the second eigenvalue. To this end, we begin by giving a necessary condition for $k(\lambda)<\infty$. This condition is stated in [JP20].

Lemma 6.1**.**

Suppose $\lambda$ is such that $k(\lambda)<\infty$. Then $\lambda$ is an algebraic integer all of whose Galois conjugates are real and at most $\lambda$ in magnitude.

Proof.

Let $G$ be any graph, and consider the characteristic polynomial $p_{G}(t)=\det(tI-A_{G})$. This polynomial is monic of degree $|V(G)|$ and has integer coefficients; this implies (1) that every root of $p_{G}$ is an algebraic integer and (2) that if $\lambda$ is a root of $p_{G}$, then every Galois conjugate of $\lambda$ is as well (the minimal polynomial of $\lambda$ must divide $p_{G}$). Since $A_{G}$ is symmetric, every eigenvalue of $A_{G}$ is real. The remainder of the lemma follows from the fact that, since every entry of $A_{G}$ is nonnegative, every eigenvalue of $A_{G}$ is at most $\lambda_{1}(A_{G})$ in magnitude, since $\lambda_{1}(A_{G})$ is a Perron eigenvalue. ∎

Remark 6.2**.**

This condition is not sufficient. For any interval $I\subset\mathbb{R}$ of length strictly exceeding $4$ (resp. strictly less than $4$), there exist infinitely many (resp. finitely many) totally real algebraic integers all of whose Galois conjugates lie within $I$ [Rob62]. In particular, there are infinitely many totally real algebraic integers all of whose conjugates lie in $[-1.995,2.006]$, and, for all but finitely many of them, the largest conjugate strictly exceeds $2$. On the other hand, due to a result of [CDG82] (see [CR90, Theorem 2.3] and surrounding remarks) no undirected graph has largest eigenvalue in the interval $(2,2.007]$.

Our choice of $\lambda$ is described in the following corollary.

Corollary 6.3**.**

If $t$ is a positive integer which is not a square, then $k(2\sqrt{t}-1)=\infty$.

Proof.

For non-square $t$, $-2\sqrt{t}-1$ is a Galois conjugate of $2\sqrt{t}-1$ and exceeds $2\sqrt{t}-1$ in magnitude. ∎

Now, we give a result which allows us to derive sets of equiangular lines in every dimension from the graphs we have constructed.

Lemma 6.4**.**

Suppose there exists an $n$-vertex $d$-regular graph $G$ with second eigenvalue $\lambda>0$ of multiplicity $k$, and let $\alpha=1/(2\lambda+1)$.

1. (a)

If $n\geq 2d$, then $N_{\alpha}(n-k)\geq n$. 2. (b)

If moreover $\lambda\geq 2\sqrt{d-1}$ and $\lambda\geq 2d/3$, then $N_{\alpha}(m)\geq m+k$ for every integer $m\geq n-k$.

Proof.

Let $A_{G}$ be the adjacency matrix of $G$; note that, since $G$ is $d$-regular, the vector $v_{1}$ with each component $1/\sqrt{n}$ is the top eigenvector of $A_{G}$ with eigenvalue $d$. Now, let $J$ be the $n\times n$ all-ones matrix, and consider the $n\times n$ symmetric matrix

[TABLE]

Since $J=nv_{1}v_{1}^{\intercal}$ and $A_{G}$ are simultaneously diagonalizable, the spectrum of $M$ consists of one copy of $1-\alpha+\alpha n-2\alpha d>0$ corresponding to the all-ones vector and a copy of $(1-\alpha)-2\alpha\mu$ for every eigenvalue $\mu<d$ of $A_{G}$, with multiplicity equal to that of $\mu$. Since $\lambda_{2}(A_{G})=\lambda=(1-\alpha)/(2\alpha)$, this means that $M$ is positive semidefinite and has eigenvalue [math] with multiplicity $k$. So, $M$ is a Gram matrix of $n$ vectors $w_{1},\dots,w_{n}\in\mathbb{R}^{n-k}$. By its definition, every diagonal entry of $M$ is $1$ and every off-diagonal entry is $\pm\alpha$. So, each $w_{i}$ is a unit vector, and the lines spanned by distinct $w_{i}$ each meet at an angle of $\arccos\alpha$. This finishes the proof of part (a).

We now show part (b). By the above argument, it suffices to show, for each integer $\ell\geq n$, the existence of a positive semidefinite $\ell\times\ell$ matrix whose diagonal consists only of ones, whose off-diagonal entries are each $\pm\alpha$, and whose rank is at most $\ell-k$. We will show that, if an $n$-vertex graph $G$ exists as in the lemma statement, such a matrix exists for each $\ell\in[n,2n)$. From here, it suffices to note that, if such an $n$-vertex graph $G=G_{0}$ exists, there is a $2^{i}n$-vertex graph $G_{i}$ also satisfying the necessary properties for all $i\geq 0$: by Theorem 2.1 and the fact that $\lambda\geq 2\sqrt{d-1}$, we can pick $G_{i+1}$ to be a suitable $2$-lift of $G_{i}$.

Now, consider the $n\times n$ matrix $M=(1-\alpha)I+\alpha J-2\alpha A_{G}$ described above; it is positive semidefinite, and the all-ones vector is an eigenvector of $M$ with eigenvalue $\alpha n-(2\alpha d-1+\alpha)$. For each integer $\ell\in(n,2n)$, define the $\ell\times\ell$ matrix $M_{\ell}$ by appending $\ell-n$ rows and columns to $M$, filling the diagonal with ones and all off-diagonal entries with $\alpha$. If $Mv=0$, then $v\oplus(0,\dots,0)$ is an eigenvector of $M_{\ell}$ with eigenvalue [math], so $M_{\ell}$ has rank at most $\ell-k$. Moreover, the diagonal entries of $M_{\ell}$ are all $1$ and the off-diagonal entries are $\pm\alpha$. So, it suffices to prove that $M_{\ell}$ is positive semidefinite. For each $r$, let $1_{r}$ be the all-ones vector of length $r$, and let $t=\ell-n$; we need to show

[TABLE]

for each $u\in\mathbb{R}^{n}$ and $v\in\mathbb{R}^{t}$. Indeed, since $1_{n}$ is an eigenvector of $M$ with eigenvalue $1-\alpha+\alpha n-2\alpha d$,

[TABLE]

where we have set $x=u^{\intercal}1_{n}$ and $y=v^{\intercal}1_{t}$, and used that $M$ is positive semidefinite. It thus suffices that

[TABLE]

The left side is decreasing in $t$, so we need only verify the inequality when $t=n$. Upon substituting $(1-\alpha)/\alpha=2\lambda$, the sufficient condition becomes

[TABLE]

This simplifies to $\frac{\lambda}{d-\lambda}-1\geq\frac{2\lambda}{n}$; the facts that $\lambda\geq 2d/3$ and $n\geq 2d\geq 2\lambda$ suffice to verify this inequality. ∎

Remark 6.5**.**

It is not generally true that $N_{\alpha}(n+1)\geq N_{\alpha}(n)+1$. In fact, $N_{1/3}(7)=\cdots=N_{1/3}(14)=28$ [GSY21].

Finally, we prove Theorem 1.4.

Proof of Theorem 1.4.

Let $u\geq 10^{5}$ be an integer, and let

[TABLE]

We will show that $n+\Omega(\log\log n)\leq N_{\alpha}(n)\leq n+O(n/\log\log n)$.

Let $\lambda=(1-\alpha)/(2\alpha)=2\sqrt{u^{2}+1}-1$. The upper bound on $N_{\alpha}(n)$ follows from Theorem 1.2 and Corollary 6.3: since $u^{2}+1$ is not a square, Corollary 6.3 gives that $k(\lambda)=\infty$, so Theorem 1.2(b) implies $N_{\alpha}(n)=n+O(n/\log\log n)$. For the lower bound, we use the graphs from Theorem 5.8. For some $d\leq 5u/2\leq 3\lambda/2$ and some sequence $n_{0}<n_{1}<\cdots$ satisfying $n_{i}\leq 4n_{i-1}^{9}$, there exist $d$-regular graphs on $n_{i}$ vertices with second eigenvalue $\lambda$ of multiplicity $\Omega(\log\log n_{i})$. Defining $f:\mathbb{N}\to\mathbb{R}$ by $f(n)=0$ for $n<n_{0}$ and $f(n)=i$ whenever $n_{i}\leq n<n_{i+1}$, we have $f(n)=\Omega(\log\log n)$. Since $\lambda\geq 2d/3$, Lemma 6.4 gives that $N_{\alpha}(n)\geq n+i$ as long as $n\geq n_{i}-i$; this implies

[TABLE]

as desired. ∎

Acknowledgment

The author would like to thank Yufei Zhao for his mentorship, for suggesting the problem, and for providing helpful comments and suggestions.