Unifying adjacency, Laplacian, and signless Laplacian theories

TL;DR

This paper introduces a unified matrix $B_\alpha(G)$ that interpolates between adjacency, Laplacian, and signless Laplacian matrices, and explores its spectral properties to unify and extend graph spectral theories.

Contribution

It defines the $B_\alpha$-matrix as a new unified framework for spectral graph theory, connecting various classical matrices and deriving new bounds and properties.

Findings

Eigenvalues of $B_\alpha(G)$ vary continuously with $\alpha$

Characterization of positive semidefinite $B_\alpha$-matrices

Bounds for eigenvalues, independence number, and chromatic number

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.