On the eigenvalues of eccentricity matrix of graphs

Abstract The eccentricity matrix E ( G ) (Randic, 2013; Wang et al., 2018) is derived from the distance matrix by keeping for each row and each column only the largest distances and leaving zeros in the remaining ones. Precisely, the element E u v in E ( G ) is defined by E u v = d ( u , v ) if d ( u , v ) = min { e ( u ) , e ( v ) } 0 otherwise. where d ( u , v ) is the distance between u and v , and e ( v ) = max { d ( u , v ) ∣ u ∈ V ( G ) } is the eccentricity of a vertex v ∈ V ( G ) . The E -eigenvalues of a graph G are those of its eccentricity matrix, and form its E -spectrum. The E -energy of G is the sum of the absolute values of E -eigenvalues. In this paper, we characterize the graphs whose second least E -eigenvalue is greater than − 15 − 193 ; moreover it is shown that all these graphs are determined by their E -spectrum. We give an upper bound for the E -energy, and discuss the corresponding extreme graphs which involve the graphs with three distinct E -eigenvalues. We finally pose some problems to be further studied, and list the E -spectra of graphs with order at most six.