On the least eigenvalue of A_α-matrix of graphs

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 A_α(G) as A_α(G)=αD(G)+(1−α)A(G), where 0≤α≤1. Then A _{[Formula presented]} (G)= [Formula presented] (D(G)+A(G))= [Formula presented] Q(G), where Q(G) is the signless Laplacian matrix of G. Let q_n(G) and λ_n(A_α) be the least eigenvalue of Q(G) and A_α(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)= [Formula presented] −1; (2) To characterize the graphs for which q_n(G)=(1− [Formula presented] ) [Formula presented]. In this paper, we present an upper bound on λ_n(A_α) 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 A_α(G). When α≠1/χ, we obtain some necessary conditions for λ_n(A_α)= [Formula presented] and, as a corollary, for the equality in problem (2).