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

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.