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 $t \geq \frac32(s-3) +\frac{4}{s-1}$.
@article{10_37236_13410,
author = {William Linz and Linyuan Lu and Zhiyu Wang},
title = {Maximum spread of {\(K_{s,t}\)-minor-free} graphs},
journal = {The electronic journal of combinatorics},
year = {2025},
volume = {32},
number = {1},
doi = {10.37236/13410},
zbl = {1561.05068},
url = {http://geodesic.mathdoc.fr/articles/10.37236/13410/}
}
TY - JOUR
AU - William Linz
AU - Linyuan Lu
AU - Zhiyu Wang
TI - Maximum spread of \(K_{s,t}\)-minor-free graphs
JO - The electronic journal of combinatorics
PY - 2025
VL - 32
IS - 1
UR - http://geodesic.mathdoc.fr/articles/10.37236/13410/
DO - 10.37236/13410
ID - 10_37236_13410
ER -
%0 Journal Article
%A William Linz
%A Linyuan Lu
%A Zhiyu Wang
%T Maximum spread of \(K_{s,t}\)-minor-free graphs
%J The electronic journal of combinatorics
%D 2025
%V 32
%N 1
%U http://geodesic.mathdoc.fr/articles/10.37236/13410/
%R 10.37236/13410
%F 10_37236_13410
William Linz; Linyuan Lu; Zhiyu Wang. Maximum spread of \(K_{s,t}\)-minor-free graphs. The electronic journal of combinatorics, Tome 32 (2025) no. 1. doi: 10.37236/13410
