The Colin de Verdi\`ere parameter, excluded minors, and the spectral radius

TL;DR

This paper characterizes graphs with maximum spectral radius constrained by the Colin de Verdière parameter and minor exclusions, revealing connections to extremal edge counts for certain minors.

Contribution

It provides a new characterization of extremal graphs maximizing spectral radius under Colin de Verdière and minor exclusion constraints.

Findings

Extremal graphs match maximum edge counts for small minors.

Characterization of graphs with maximum spectral radius for given parameters.

Differences in extremal structures when minors are larger.

Abstract

In this paper we characterize graphs which maximize the spectral radius of their adjacency matrix over all graphs of Colin de Verdi\`ere parameter at most $m$. We also characterize graphs of maximum spectral radius with no $H$ as a minor when $H$ is either $K_r$ or $K_{s,t}$. Interestingly, the extremal graphs match those which maximize the number of edges over all graphs with no $H$ as a minor when $r$ and $s$ are small, but not when they are larger.