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 Aₐ(G), defined for any real α ∈ [0, 1], as: Aₐ(G) = α D(G) + (1 - α) A(G), and also mentioned that the matrices Aₐ can underpin a unified theory of A(G) and Q(G). Inspired by this definition, we define a family of Bₐ-matrices of G, denoted by: Bₐ(G) = α A(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), and A(G) = B₁(G). In this article, we study several spectral properties of Bₐ-matrices to unify the theories of adjacency, Laplacian, and signless Laplacian matrices of graphs. In particular, we prove that each eigenvalue of Bₐ(G) is continuous with respect to α. Using this, we characterize positive semidefinite Bₐ-matrices in terms of α. As a consequence, we provide an upper bound on the independence number of G. Moreover, we establish some bounds for the largest and smallest eigenvalues of Bₐ(G). As a result, we obtain a bound for the chromatic number of G and deduce several known results. Additionally, we present a Sachs-type result for the characteristic polynomial of a Bₐ-matrix.