Unifying adjacency, Laplacian, and signless Laplacian theories

Abstract

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 $\alpha \in [0,1]$ 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 $\alpha \in [0,1]$. Note that $L(G)=B_0(G),D(G)=2B_{\frac{1}{2}}(G),Q(G)=3B_{\frac{2}{3}}(G),A(G)=B_1(G)$. In this article, we study several spectral properties of $B_{\alpha }$-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.

The final version of the article to be appear in Ars Mathematica Contemporanea