Spectral extrema of $K_{s,t}$-minor free graphs--On a conjecture of M. Tait

TL;DR

This paper fully resolves Tait's conjecture on the maximum spectral radius of $K_{s,t}$-minor free graphs and characterizes the extremal graphs, also extending results to $K_{1,t}$-minor free graphs using advanced spectral and structural methods.

Contribution

It provides a complete proof of Tait's conjecture and determines extremal graphs for $K_{s,t}$-minor and $K_{1,t}$-minor free graphs, introducing new spectral tools.

Findings

Confirmed Tait's conjecture for all sufficiently large $n$.

Identified the unique extremal graphs achieving maximum spectral radius.

Extended spectral extremal results to $K_{1,t}$-minor free graphs.

Abstract

Minors play an important role in extremal graph theory and spectral extremal graph theory. Tait [The Colin de Verdi\`{e}re parameter, excluded minors, and the spectral radius, J. Combin. Theory Ser. A 166 (2019) 42--58] determined the maximum spectral radius and characterized the unique extremal graph for $K_r$-minor free graphs of sufficiently large order $n$, he also made great progress on $K_{s,t}$-minor free graphs and posed a conjecture: Let $2\leq s\leq t$ and $n-s+1=pt+q$, where $n$ is sufficiently large and $1\leq q\leq t.$ Then $K_{s-1}\nabla (pK_t\cup K_q)$ is the unique extremal graph with the maximum spectral radius over all $n$-vertex $K_{s,t}$-minor free graphs. In this paper, Tait's conjecture is completely solved. We also determine the maximum spectral radius and its extremal graphs for $n$-vertex $K_{1,t}$-minor free graphs. To prove our results, some spectral and structural tools, such as, local edge maximality, local degree sequence majorization, double eigenvectors transformation, are used to deduce structural properties of extremal graphs.