On the α-index of graphs with pendent paths

Let G be a graph with adjacency matrix A(G) and let D(G) be the diagonal matrix of the degrees of G. For every real alpha is an element of [0,1], write A(alpha) (G) for the matrix A(alpha) (G) = alpha D (G) + (1 - alpha)A (G). This paper presents some extremal results about the spectral radius rho(alpha) (G) of A(alpha) (G) that generalize previous results about (rho 0) (G) and rho(1/2) (G). In particular, write B-p,B-q,B-r be the graph obtained from a complete graph K-p by deleting an edge and attaching paths P-q and P-r to its ends. It is shown that if alpha is an element of [0,1) and G is a graph of order n and diameter at least k, then rho alpha (G) <= rho alpha(B-n-k+2,B-[k/2],B-[k/2]) with equality holding if and only if G = B-n-k+2,B-[k/2],B-[k/2]. This result generalizes results of Hansen and Stevanovic [5], and Liu and Lu [7]. (c) 2018 Elsevier Inc. All rights reserved.