On the least eigenvalue of Aα-matrix of graphs

Abstract

Let G be a graph of order n with m edges and chromatic number χ. Let A(G) be the adjacency matrix and D(G) be the diagonal matrix of vertex degrees of G. Nikiforov defined the matrix Aα(G) as Aα(G)=αD(G)+(1−α)A(G), where 0≤α≤1. Then A12(G)=12(D(G)+A(G))=12Q(G), where Q(G) is the signless Laplacian matrix of G. Let qn(G) and λn(Aα) be the least eigenvalue of Q(G) and Aα(G), respectively. Lima et al. (2011) [8] proposed some open problems on qn(G), two of which are as follows:(1) To characterize the graphs for which qn(G)=2mn−1;(2) To characterize the graphs for which qn(G)=(1−1χ−1)2mn.In this paper, we present an upper bound on λn(Aα) in terms of n, m and α (1/2≤α≤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 Aα(G). When α≠1/χ, we obtain some necessary conditions for λn(Aα)=(αχ−1)2m(χ−1)n and, as a corollary, for the equality in problem (2).