Maximum spread of graphs and bipartite graphs

TL;DR

This paper proves two longstanding conjectures about the maximum eigenvalue spread in graphs, identifying extremal structures for large graphs and bipartite graphs with fixed edges, using advanced mathematical techniques.

Contribution

It resolves two 20-year-old conjectures on graph eigenvalue spread, establishing extremal graph structures for large graphs and fixed edge counts.

Findings

The maximum spread graph for large n is the join of a clique and an independent set.

Asymptotic proof that maximum spread graphs with fixed edges are bipartite.

Counterexamples show the asymptotic results are tight.

Abstract

Given any graph $G$, the (adjacency) spread of $G$ is the maximum absolute difference between any two eigenvalues of the adjacency matrix of $G$. In this paper, we resolve a pair of 20-year-old conjectures of Gregory, Hershkowitz, and Kirkland regarding the spread of graphs. The first states that for all positive integers $n$, the $n$-vertex graph $G$ that maximizes spread is the join of a clique and an independent set, with $\lfloor 2n/3 \rfloor$ and $\lceil n/3 \rceil$ vertices, respectively. Using techniques from the theory of graph limits and numerical analysis, we prove this claim for all $n$ sufficiently large. As an intermediate step, we prove an analogous result for a family of operators in the Hilbert space over $\mathscr{L}^2[0,1]$. The second conjecture claims that for any fixed $e\leq n^2/4$, if $G$ maximizes spread over all $n$-vertex graphs with $e$ edges, then $G$ is bipartite. We prove an asymptotic version of this conjecture. Furthermore, we exhibit an infinite family of counterexamples, which shows that our asymptotic solution is tight up to lower order error terms.