Maximum spread of $K_{s,t}$-minor-free graphs
William Linz, Linyuan Lu, Zhiyu Wang
TL;DR
The paper analyzes the maximum spread $S(G)$ over the family of $K_{s,t}$-minor-free graphs on $n$ vertices for large $n$, building on spectral extremal methods. It proves that the extremal structure is a join between a fixed $(s-1)$-vertex graph $L_{ ext{max}}$ and a cluster of $t$-cliques together with isolated vertices: $G=L_{ ext{max}} \\vee (\\ell_0 K_t \\cup (n-s+1 - t\\ell_0) P_1)$, where $L_{ ext{max}}$ maximizes a potential function $\\psi(L)$ and $\\ell_0$ is chosen near $\\ell_1=\\frac{2}{3t}(n-s+1-\\frac{(t-1)^2}{9(s-1)})$. They derive an asymptotic expansion for the maximum spread in terms of $a_0=(s-1)(n-s+1)$ and $\\psi(L_{ ext{max}})$, and identify an admissibility criterion $\\psi(L) \\le 0$ that yields a threshold on $t$: $t \\ge \\ rac{3}{2}(s-3) + \\frac{4}{s-1}$. The results provide explicit structure and near-optimal spread values for admissible pairs $(s,t)$, with a plan to address non-admissible cases in a sequel.
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 $ξ_{t}$ 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 $\lfloor \frac{2n+ξ_{t}}{3t}\rfloor$ copies of $K_t$ and $n-s+1 - t\lfloor \frac{2n+ξ_t}{3t}\rfloor$ isolated vertices. Furthermore, we give an explicit formula for $ξ_{t}$ and an explicit description for the graph $L$ for $t \geq \frac32(s-3) +\frac{4}{s-1}$.