Spectral extrema of Ks, t-minor free graphs - On a conjecture of M. Tait

Minors play an important role in extremal graph theory and spectral extremal graph theory. Tait [The Colin de Verdière 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 Kr-minor free graphs of sufficiently large order n, he also made great progress onKs,t-minor free graphs and posed a conjecture: Let 2 ≤ s ≤ t and n − s + 1 = pt + q, where n is sufficiently large and 1 ≤ q ≤ t. Then Ks−1∇(pKt∪Kq) is the unique extremal graph with the maximum spectral radius over all n-vertex Ks,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 K1,tminor 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.

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