Maximum spectral gaps of graphs

TL;DR

This paper investigates the maximum spectral gaps between eigenvalues of graphs, providing bounds, identifying extremal graphs for specific pairs, and extending previous work on the spread of graphs.

Contribution

It introduces bounds for eigenvalue differences for all pairs (i,j), identifies cases of tight bounds, and proves the uniqueness of extremal graphs for the pair (1,0).

Findings

Provided upper bounds for eigenvalue differences for all (i,j) pairs.

Identified an infinite family of pairs where bounds are tight.

Proved the extremal graph for (1,0) is unique.

Abstract

The spread of a graph $G$ is the difference $\lambda_1 - \lambda_n$ 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 $\lambda_{i+1} - \lambda_{n-j}$ 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.