On the Eccentricity Laplacian and Eccentricity Signless Laplacian Matrices of a Graph

TL;DR

This paper introduces new Laplacian matrices based on eccentricity for connected graphs, exploring their spectral properties and characterizations of specific graph classes.

Contribution

It defines eccentricity Laplacian and signless Laplacian matrices and establishes their spectral equivalences and characterizations for certain graph classes.

Findings

Spectral characterization of $\\mathcal{E}$-bipartite graphs.

Symmetry of the $\\mathcal{E}$-spectrum indicates graph properties.

Equivalence among different eccentricity-based matrices for various graphs.

Abstract

In this paper, we introduce the Laplacian and the signless Laplacian for the eccentricity matrix of a connected graph, referred to as the eccentricity Laplacian and the eccentricity signless Laplacian, respectively. We establish the equivalence among the eccentricity Laplacian, eccentricity signless Laplacian, and eccentricity spectrum for different classes of graphs. We provide spectral characterization of $\mathcal{E}$-bipartite graphs by the symmetry of $\mathcal{E}$-spectrum and the similarity of these Laplacian matrices.