Unifying adjacency, Laplacian, and signless Laplacian theories*

Article Type

Research Article

Publication Title

Ars Mathematica Contemporanea

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 Aα(G) defined for any real α ∈ [0, 1] as Aα(G):= α D(G) + (1 − α) A(G), and also mentioned that the matrices Aα(G) can underpin a unified theory of A(G) and Q(G). Inspired from the above definition, we introduce the Bα-matrix of G, Bα(G):= αA(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 Bα-matrices to unify the theories of adjacency, Laplacian, and signless Laplacian matrices of graphs. In particular, we prove that each eigenvalue of Bα(G) is continuous on α. Using this, we characterize positive semidefinite Bα-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 Bα(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 Bα-matrix.

DOI

10.26493/1855-3974.3163.6hw

Publication Date

1-1-2024

Comments

Open Access; Green Open Access; Hybrid Gold Open Access

Link to Full Text

Share

COinS