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):=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), 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.