Maximum spread of $K_r$-minor free graphs

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.