Maximum Spread of $K_{s,t}$-Minor-Free Graphs

• Published in: The Electronic journal of combinatorics Vol. 32; no. 1
• Main Authors: Linz, William, Lu, Linyuan, Wang, Zhiyu
• Format: Journal Article
• Language: English
• Published: 17.01.2025
• ISSN: 1077-8926; 1077-8926
• DOI: 10.37236/13410

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 $t \geq \frac32(s-3) +\frac{4}{s-1}$.