Maximum spread of $K_{s,t}$-minor-free graphs

TL;DR

This paper determines the maximum spread of large $K_{s,t}$-minor-free graphs and characterizes the extremal structures, providing explicit formulas and descriptions for the extremal graphs based on parameters $s$ and $t$.

Contribution

It establishes the extremal graphs that maximize spread among $K_{s,t}$-minor-free graphs for large $n$, including explicit formulas and structural descriptions.

Findings

Identifies extremal graphs for maximum spread in $K_{s,t}$-minor-free family.

Provides explicit formula for the parameter $\xi_t$.

Characterizes the structure of extremal graphs for large $n$.

Abstract

The spread of a graph $G$ is the difference between the largest and smallest eigenvalue of the adjacency matrix of $G$. In this paper, we consider the family of graphs which contain no $K_{s,t}$-minor. We show that for any $t\geq s \geq 2$ and sufficiently large $n$, there is an integer $\xi_{t}$ such that the extremal $n$-vertex $K_{s,t}$-minor-free graph attaining the maximum spread is the graph obtained by joining a graph $L$ on $(s-1)$ vertices to the disjoint union of $\lfloor \frac{2n+\xi_{t}}{3t}\rfloor$ copies of $K_t$ and $n-s+1 - t\lfloor \frac{2n+\xi_t}{3t}\rfloor$ isolated vertices. Furthermore, we give an explicit formula for $\xi_{t}$ and an explicit description for the graph $L$ for $t \geq \frac32(s-3) +\frac{4}{s-1}$.