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

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