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_α(G) defined for any real α ∈ [0, 1] as A_α(G):= α D(G) + (1 − α) A(G), and also mentioned that the matrices A_α(G) can underpin a unified theory of A(G) and Q(G). Inspired from the above definition, we introduce the B_α-matrix of G, B_α(G):= αA(G) + (1 − α)L(G) for α ∈ [0, 1]. Note that L(G) = B₀(G), D(G) = 2B1 ₂ (G), Q(G) = 3B2 ₃ (G), A(G) = B₁(G). In this article, we study several spectral properties of B_α-matrices to unify the theories of adjacency, Laplacian, and signless Laplacian matrices of graphs. In particular, we prove that each eigenvalue of B_α(G) is continuous on α. Using this, we characterize positive semidefinite B_α-matrices in terms of α. 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_α(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_α-matrix.