Maximum spread of $K_{2,t}$-minor-free graphs

TL;DR

This paper determines the maximum eigenvalue spread of $K_{2,t}$-minor-free graphs, identifying the extremal structures and providing an explicit formula for a key parameter.

Contribution

It characterizes the extremal graphs achieving maximum spread in $K_{2,t}$-minor-free graphs and derives an explicit formula for the parameter $\xi_t$.

Findings

Maximum spread achieved by specific join graphs involving $K_t$ and isolated vertices.

Unique extremal graph structure except for certain modular cases.

Explicit formula for the parameter $\xi_t$.

Abstract

The spread of a graph $G$ is the difference between the largest and smallest eigenvalues of the adjacency matrix of $G$. In this paper, we consider the family of graphs which contain no $K_{2,t}$-minor. We show that for any $t\geq 2$, there is an integer $\xi_t$ such that the maximum spread of an $n$-vertex $K_{2,t}$-minor-free graph is achieved by the graph obtained by joining a vertex to the disjoint union of $\lfloor \frac{2n+\xi_t}{3t}\rfloor$ copies of $K_t$ and $n-1 - t\lfloor \frac{2n+\xi_t}{3t}\rfloor$ isolated vertices. The extremal graph is unique, except when $t\equiv 4 \mod 12$ and $\frac{2n+ \xi_t} {3t}$ is an integer, in which case the other extremal graph is the graph obtained by joining a vertex to the disjoint union of $\lfloor \frac{2n+\xi_t}{3t}\rfloor-1$ copies of $K_t$ and $n-1-t(\lfloor \frac{2n+\xi_t}{3t}\rfloor-1)$ isolated vertices. Furthermore, we give an explicit formula for $\xi_t$.