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Maximum spectral gaps of graphs

George Brooks, William Linz, Linyuan Lu

TL;DR

Given a graph $G$, the authors study the maximization of the $(i,j)$-spread $spread_{i,j}(G)=λ_{i+1}-λ_{n-j}$ over graphs on $n$ vertices, and the asymptotic constant $s_{i,j}=\lim_{n→∞} spread_{i,j}(n)/n$. They extend the problem to the larger class $L_n$ of graphs with at most one self-loop per vertex and prove $s_{i,j}^*=s_{i,j}$, enabling uniform upper bounds across both simple and looped graphs. They obtain universal bounds such as $λ_{i+1}-λ_{n-j} ≤ \tfrac{n}{2} \sqrt{\tfrac{i+j+1}{i(j+1)}}$ for all $i,j$, and a sharper bound for the first eigenvalue, $λ_1-λ_{n-j} ≤ \tfrac{n}{2}\left(1+\sqrt{\tfrac{j+2}{j+1}}\right)$, with infinitely many cases where these are tight; they also show $λ_{k+1}-λ_{n-k+1} ≤ \tfrac{n}{\sqrt{2k}}$ when equality occurs if a symmetric Hadamard matrix of order $2k$ exists. The extremal constructions involve blowups of small seed graphs such as $P^*_4$ and, in the tight $k$-Hadamard cases, specific Kronecker-product graphs, with the $k=1$ case proved to be unique via a spectral-structure argument. The section connects these findings to Nikiforov's broader program of maximizing linear combinations of eigenvalues and presents concrete lower bounds and conjectures guiding future work.

Abstract

The spread of a graph $G$ is the difference $λ_1 - λ_n$ between the largest and smallest eigenvalues of its adjacency matrix. Breen, Riasanovsky, Tait and Urschel recently determined the graph on $n$ vertices with maximum spread for sufficiently large $n$. In this paper, we study a related question of maximizing the difference $λ_{i+1} - λ_{n-j}$ for a given pair $(i, j)$ over all graphs on $n$ vertices. We give upper bounds for all pairs $(i, j)$, exhibit an infinite family of pairs where the bound is tight, and show that for the pair $(1, 0)$ the extremal example is unique. These results contribute to a line of inquiry pioneered by Nikiforov aiming to maximize different linear combinations of eigenvalues over all graphs on $n$ vertices.

Maximum spectral gaps of graphs

TL;DR

Given a graph , the authors study the maximization of the -spread over graphs on vertices, and the asymptotic constant . They extend the problem to the larger class of graphs with at most one self-loop per vertex and prove , enabling uniform upper bounds across both simple and looped graphs. They obtain universal bounds such as for all , and a sharper bound for the first eigenvalue, , with infinitely many cases where these are tight; they also show when equality occurs if a symmetric Hadamard matrix of order exists. The extremal constructions involve blowups of small seed graphs such as and, in the tight -Hadamard cases, specific Kronecker-product graphs, with the case proved to be unique via a spectral-structure argument. The section connects these findings to Nikiforov's broader program of maximizing linear combinations of eigenvalues and presents concrete lower bounds and conjectures guiding future work.

Abstract

The spread of a graph is the difference between the largest and smallest eigenvalues of its adjacency matrix. Breen, Riasanovsky, Tait and Urschel recently determined the graph on vertices with maximum spread for sufficiently large . In this paper, we study a related question of maximizing the difference for a given pair over all graphs on vertices. We give upper bounds for all pairs , exhibit an infinite family of pairs where the bound is tight, and show that for the pair the extremal example is unique. These results contribute to a line of inquiry pioneered by Nikiforov aiming to maximize different linear combinations of eigenvalues over all graphs on vertices.
Paper Structure (4 sections, 7 theorems, 60 equations, 5 figures, 3 tables)

This paper contains 4 sections, 7 theorems, 60 equations, 5 figures, 3 tables.

Table of Contents

  1. Introduction
  2. Notations and Lemmas
  3. Proofs of the theorems
  4. Lower bounds and conjectures

Key Result

Theorem 1

For any positive integer $i$ and nonnegative integer $j$ with $i, j \le n$, for all graphs $G^*$ on $n$ vertices with at most one loop per vertex, we have Thus,

Figures (5)

  • Figure 1: Closed path $P^*_4$.
  • Figure 2: Closed cube $Q^*_3$.
  • Figure 3: The graph $K_{5}\cup K_{4}^*$.
  • Figure 4: The graph $K_{(5)*,4}$.
  • Figure 5: The graph $K_{8}^{4*}$.

Theorems & Definitions (18)

  • Theorem 1
  • Remark 1
  • Theorem 2
  • Theorem 3
  • Theorem 4
  • Remark 2
  • Lemma 1: Eigenvalues of the blowup graph
  • proof
  • Lemma 2
  • proof
  • ...and 8 more