Unifying adjacency, Laplacian, and signless Laplacian theories

Let G be a simple graph with associated diagonal matrix of vertex degrees D(G), adjacency matrix A(G), Laplacian matrix L(G) and signless Laplacian matrix Q(G). Recently, Nikiforov proposed the family of matrices A(alpha) (G) defined for any real alpha is an element of [0, 1] as A(alpha )(G) := alpha D (G) + (1 - alpha) A (G) , and also mentioned that the matrices A(alpha) (G) can underpin a unified theory of A (G) and Q (G). Inspired from the above definition, we introduce the B-alpha-matrix of G , B-alpha (G) := alpha A(G) + (1 - alpha) L(G) for alpha is an element of [0, 1] . Note that L(G) = B-0 (G) , D(G) = 2B(1/2 )(G), Q(G) = 3B2(/3) (G), A(G) = B-1 (G). In this article, we study several spectral properties of B c-matrices to unify the theories of adjacency, Laplacian, and signless Laplacian matrices of graphs. In particular, we prove that each eigenvalue of B-alpha (G) is continuous on alpha . Using this, we characterize positive semidefinite B-alpha-matrices in terms of alpha . As a consequence, we provide an upper bound of the independence number of G . Besides, we establish some bounds for the largest and the smallest eigenvalues of B-alpha (G) . As a result, we obtain a bound for the chromatic number of G and deduce several known results. In addition, we present a Sachs-type result for the characteristic polynomial of a B-alpha-matrix.