The extremal α-index of outerplanar and planar graphs

Abstract

Let G be a graph of order n with adjacency matrix A(G) and let D(G) be the diagonal matrix of the degrees of G. For any real α ∈ [0, 1], write Aα(G) for the matrixAα(G)=αD(G)+(1−α)A(G).This paper shows some extremal results on the spectral radius ρα(G) of Aα(G). We determine the upper bound on ρα(G) if α ∈ (0, 1) and G is a graph with no K2, t (t ≥ 3) minor. We also show that the unique outerplanar graph of order n with maximum ρα(G) is the join of a vertex and a path Pn−1. Moreover, we prove that the unique planar graph of order n with maximum signless Laplacian spectral radius is the join of an edge and a path Pn−2.