2019/09/12 by Mahato, Iswar, Gurusamy, R., Kannan, M. Rajesh +1
#Combinatorics (math.CO) #FOS: Mathematics
paper · doi:10.48550/arxiv.1909.05609
The eccentricity matrix ε(G) of a graph G is obtained from the distance matrix by retaining the eccentricities (the largest distance) in each row and each column. In this paper, we give a characterization of the star graph, among the trees, in terms of invertibility of the associated eccentricity matrix. The largest eigenvalue of ε(G) is called the ε-spectral radius, and the eccentricity energy (or the ε-energy) of G is the sum of the absolute values of the eigenvalues of ε(G). We establish some bounds for the ε-spectral radius and characterize the extreme graphs. Two graphs are said to be ε-equienergetic if they have the same ε-energy. For any n ≥ 5, we construct a pair of ε-equienergetic graphs on n vertices, which are not ε-cospectral.