Maximum spread of $K_r$-minor free graphs

TL;DR

This paper characterizes the maximum eigenvalue spread in large graphs that do not contain a complete graph minor, identifying the extremal structure as a join of a clique and an independent set.

Contribution

It provides a precise extremal characterization of the maximum spread for $K_r$-minor free graphs, a problem previously unexplored.

Findings

Maximum spread graphs are joins of a clique and an independent set.

The extremal structure is valid for sufficiently large $n$.

The maximum spread is achieved by a specific join configuration.

Abstract

The spread of a graph is the difference between the largest and smallest eigenvalue of its adjacency matrix. In this paper, we investigate spread problems for graphs with excluded clique-minors. We show that for sufficiently large $n$, the $n$-vertex $K_r$-minor free graph with maximum spread is the join of a clique and an independent set, with $r-2$ and $n-r+2$ vertices, respectively.