On the least eigenvalue of Aα-matrix of graphs

Let G be a graph of order n with m edges and chromatic number chi. Let A(G) be the adjacency matrix and D(G) be the diagonal matrix of vertex degrees of G. Nikiforov defined the matrix A(alpha)(G) as A(alpha)(G) = alpha D(G) + (1 - alpha)A(G), where 0 <= alpha <= 1. Then A(1/2) (G) = 1/2 (D(G)+ A(G)) = 1/2 Q(G), where Q(G) is the signless Laplacian matrix of G. Let q(n)(G) and lambda(n)(A(alpha)) be the least eigenvalue of Q(G) and A(alpha)(G), respectively. Lima et al. (2011) [8] proposed some open problems on q(n)(G), two of which are as follows: (1) To characterize the graphs for which q(n)(G) = 2m/n -1; (2) To characterize the graphs for which q(n)(G) = (1 - 1/chi - 1) x 2m/n. In this paper, we present an upper bound on lambda(n) (A(alpha)) in terms of n, m and alpha (1/2 <= alpha <= 1), and characterize the extremal graphs. As an application, we completely solve problem (1). Moreover, we examine the more generalized result of problem (2) on A(alpha)(G). When alpha not equal 1/chi, we obtain some necessary conditions for lambda(n)(A(alpha)) = (alpha(chi)-1)2m/(chi-1)(n) and, as a corollary, for the equality in problem (2). (C) 2019 Elsevier Inc. All rights reserved.