Some Remarks on Systems of Equiangular Lines

TL;DR

This paper investigates the maximum size of equiangular line systems in real space with a specific cosine condition, motivated by prior work and focusing on cases where the reciprocal of the cosine is not an odd positive integer.

Contribution

It provides new insights into the bounds of equiangular line systems under certain cosine constraints, extending previous results in the field.

Findings

Established bounds for N_α(d) when 1/α is not an odd positive integer

Connected the problem to prior conjectures and results in equiangular lines

Clarified conditions under which maximum configurations are achieved

Abstract

In this note, we study the maximum number $N_\alpha(d)$ of a system of equiangular lines in $\mathbb{R}^d$ with cosine $\alpha$, where $\frac{1}{\alpha}$ is not an odd positive integer. This note is motivated by a remark in a $2018$ paper by Balla, Dr\"{a}xler, Keevash and Sudakov.

Equations

$$\left[\begin{array}{cccc}-2A(H_1)-\mathbf{I}& & & \\ & -2A(H_2)-\mathbf{I}& & \\ & & \ddots & \\ & & & -2A(H_t)-\mathbf{I}\end{array}\right].$$

Full text

key words: Equiangular lines, Seidel matrix, smallest eigenvalue, spectral radius, rank.

††2010 Mathematics Subject Classification. Primary 05C50, secondary 05C22.††E-mail addresses: cmy1325@163.com (M.Y. Cao), koolen@ustc.edu.cn (J.H. Koolen), piez@naver.com (J.Y. Yang).

1 Introduction

A system of lines through the origin in $d$-dimensional Euclidean space $\mathbb{R}^{d}$ is called equiangular if the angle between any pair of lines is the same. Let $N_{\alpha}(d)$ be the maximum number of a system of equiangular lines in $\mathbb{R}^{d}$ with common angle $\arccos\alpha$. For more information on systems of equiangular lines, see [6]. In 2018, Balla, Dräxler, Keevash and Sudakov [1] conjectured that if $\alpha=\frac{1}{2r-1}$ for a positive integer $r\geq 2$, then $N_{\frac{1}{2r-1}}(d)=\frac{r(d-1)}{r-1}+O(1)$, for sufficiently large $d$. They also wrote ‘If $\alpha$ is not of the above form, the situation is less clear but it is conceivable that $N_{\alpha}(d)=(1+o(1))d$.’.

In this note, we will consider $N_{\alpha}(d)$ when $\frac{1}{\alpha}$ is not an odd integer. We prove the following:

Theorem 1.1**.**

Let $0<\tau\leq\frac{1}{1+2\sqrt{2+\sqrt{5}}}$. Then there exists a sequence $(\alpha_{i})^{\infty}_{i=1}$ such that

1. (i)

$\underset{i\rightarrow\infty}{\lim}{\alpha_{i}}=\tau$, 2. (ii)

for all $i$, there exists $\eta_{i}>0$ such that $N_{\alpha_{i}}(d)\geq(1+\eta_{i})d$.

2 Seidel matrices

First, we introduce Seidel matrix which is our main tool. All graphs are simple and undirected. For undefined terminologies, we refer to [4, 2].

Let $G$ be a graph with $n$ vertices. The adjacency matrix $A(G)$ of $G$ is an $n\times n$ matrix whose rows and columns are indexed by the vertices of $G$ such that $A(G)_{xy}$ is $1$ if $x$ and $y$ are adjacent vertices and [math] otherwise. The Seidel matrix $S(G)$ of $G$ is the matrix $S(G)=\mathbf{J-I}-A(G)$, where $\mathbf{J}$ is the all-ones matrix and $\mathbf{I}$ is the identity matrix. Let $\mathbf{j}$ denote the all-ones vector. The spectral radius $\rho(G)$ of a graph $G$ is the largest eigenvalue of $A(G)$.

Seidel matrices and systems of equiangular lines, are related as follow (see for example, [4, Section 11.1]):

Proposition 2.1**.**

There exists a system of $n$ equiangular lines in $\mathbb{R}^{d}$ with common angle $\arccos\alpha$ if and only if there exists a graph $G$ with $n$ vertices such that $S(G)$ has smallest eigenvalue at least $-\frac{1}{\alpha}$ and rk$(S(G)+\frac{1}{\alpha}\mathbf{I})=d$.

This leads to the following definition. The number $R_{\beta}(n)$ is defined as $R_{\beta}(n):=\min\{\textrm{rk}(S(G)+\beta\mathbf{I})\mid G$ is a graph on $n$ vertices and $S(G)$ has smallest eigenvalue at least $-\beta\}$. We obtain the following lemma immediately from Proposition 2.1.

Lemma 2.2**.**

$N_{\alpha}(d)\geq n$* if and only if $R_{\frac{1}{\alpha}}(n)\leq d.$*

Example 2.3**.**

$N_{\frac{1}{3}}(d)=28$* for $7\leq d\leq 13$ and $R_{3}(n)=7$ for $17\leq n\leq 28$ (see [6, Table $1$]).*

Now we discuss some properties of $R_{\beta}(n)$. Since $S(G)$ is a real symmetric matrix, it is diagonalizable and all its eigenvalues are totally real algebraic integers. This implies that if $\beta>1$ is not a totally real algebraic integer, then $R_{\beta}(n)=n$ and $N_{\frac{1}{\beta}}(d)=d$ for all integers $n\geq 2$ and $d\geq 2$.

Haemers observed that for a graph $G$ of order $n$, we have $\det(-S(G)+x\mathbf{I})\equiv\det(x\mathbf{I}-\mathbf{J}+\mathbf{I})\ (\rm{mod}\ 2)$ (see [5]). This shows that for example $R_{\sqrt{2}}(n)=n$ and $N_{\frac{1}{\sqrt{2}}}(d)=d$ for all integers $d\geq 2$ and $n\geq 2$. Similar but more complicated formulas for the coefficients of $\det(-S(G)+x\mathbf{I})$ are known (see for example, [5, 6, 7]). The observation of Haemers also implies that $2t+1\geq R_{2m}(2t+1)\geq 2t=R_{2m}(2t)$ and $N_{\frac{1}{2m}}(2t+1)=2t+1\geq N_{\frac{1}{2m}}(2t)\geq 2t$, where $t>m$ are positive integers. Now, we give an example of the case $R_{2m}(2t+1)=2t$ and $N_{\frac{1}{2m}}(2t)=2t+1$ for certain $m$ and $t$. Let $G$ be a $3$-regular graph on $4n^{\prime}+2$ vertices, where $n^{\prime}\geq 2$ is an integer. Then the Seidel matrix of the complement of the line graph of $G$ has smallest eigenvalue $6-6n^{\prime}$ with multiplicity one. This shows $R_{6n^{\prime}-6}(6n^{\prime}+3)=6n^{\prime}+2$ and $N_{\frac{1}{6n^{\prime}-6}}(6n^{\prime}+2)=6n^{\prime}+3$.

Next, we show the following lemma on Seidel matrices.

Lemma 2.4**.**

Let $G$ be a graph of order $n$ with $t$ connected components $H_{1},H_{2},\ldots,H_{t}$. If $\rho(H_{i})=\rho$ for all $i$, then $S(G)$ has smallest eigenvalue $-2\rho-1$ with multiplicity at least $t-1$.

Proof.

Let $B(G)=S(G)-\mathbf{J}$. We can check that $B(G)$ is the block diagonal matrix of the form

[TABLE]

The matrix $B(G)$ has smallest eigenvalue $-2\rho-1$ with multiplicity $t$, so there are $t-1$ linearly independent eigenvectors $\mathbf{v}_{1},\ldots,\mathbf{v}_{t-1}$ of $B(G)$ such that $B(G)\mathbf{v}_{i}=-2\rho-1$ and $\langle\mathbf{v}_{i},\mathbf{j}\rangle=0$, for $i=1,\ldots,t-1$. It follows that $S(G)$ has eigenvalue $-2\rho-1$ with multiplicity at least $t-1$. This shows the lemma. ∎

To show Theorem 1.1, we need the following result of Shearer [8] on spectral radius of graphs at least $\sqrt{2+\sqrt{5}}$.

Theorem 2.5**.**

For any real number $\lambda\geq\sqrt{2+\sqrt{5}}$, there exists a sequence of graphs $(G_{i})^{\infty}_{i=1}$ such that $\underset{i\rightarrow\infty}{\lim}\rho(G_{i})=\lambda$.

For the classification of the graphs with spectral radius less than $\sqrt{2+\sqrt{5}}$, see [3].

Now, we will prove the following theorem.

Theorem 2.6**.**

For a real number $\mu\geq 1+2\sqrt{2+\sqrt{5}}$, there exists a sequence $(\beta_{i})^{\infty}_{i=1}$ such that the following hold:

1. (i)

$\underset{i\rightarrow\infty}{\lim}{\beta_{i}}=\mu$, 2. (ii)

for all $i$, there exists $\varepsilon_{i}>0$ such that $R_{\beta_{i}}(n)\leq(1-\varepsilon_{i})n$.

Proof.

Let $\mu$ be a real number at least $1+2\sqrt{2+\sqrt{5}}$. For $\lambda:=\frac{\mu-1}{2}\geq\sqrt{2+\sqrt{5}}$, there exists a sequence of graphs $(G_{i})^{\infty}_{i=1}$ such that $\underset{i\rightarrow\infty}{\lim}\rho(G_{i})=\lambda$ by Theorem 2.5. Let $tG_{i}$ be the disjoint union of $t$ copies of $G_{i}$ for a positive integer $t$. For real numbers $\beta_{i}=2\rho(G_{i})+1$ and $n_{i}=n(G_{i})$, $tG_{i}$ is a graph with $tn_{i}$ vertices and the smallest eigenvalue of $S(tG_{i})$ is $-\beta_{i}$ with multiplicity at least $t-1$ by Lemma 2.4. It follows that $R_{\beta_{i}}(tn_{i})\leq tn_{i}-t+1$ by Proposition 2.1. Since $\underset{i\rightarrow\infty}{\lim}{\beta_{i}}=\mu$, this finishes the proof. ∎

Theorem 1.1 follows now from Theorem 2.6 and Lemma 2.2.

Acknowledgments

M.Y. Cao is partially supported by the National Natural Science Foundation of China (No. $11571044$ and No. $61373021$) and the Fundamental Research Funds for the Central Universities.

J.H. Koolen is partially supported by the National Natural Science Foundation of China (No. $11471009$ and No. $11671376$) and Anhui Initiative of Quantum Information Technologies (No. AHY $150000$).

J.Y. Yang is partially supported by the National Natural Science Foundation of China (No. $11371028$).