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_ (G) defined for any real 0, 1 as A_ (G): =\, D (G) + (1-) \, A (G), and also mentioned that the matrices A_ (G) can underpin a unified theory of A (G) and Q (G). Inspired from the above definition, we introduce the B_-matrix of G, B_ (G): = A (G) + (1-) L (G) for 0, 1. Note that L (G) =B₀ (G), D (G) =2B₁₂ (G), Q (G) =3B₂₃ (G), A (G) =B₁ (G). In this article, we study several spectral properties of B_ -matrices to unify the theories of adjacency, Laplacian, and signless Laplacian matrices of graphs. In particular, we prove that each eigenvalue of B_ (G) is continuous on. Using this, we characterize positive semidefinite B_ -matrices in terms of. 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_ (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_ -matrix.