Characterizing the extremal graphs with respect to the eccentricity spectral radius, and beyond

Abstract

The eccentricity matrix E(G) of a graph G is derived from the corresponding distance matrix by keeping only the largest non-zero elements for each row and each column and leaving zeros for the remaining ones. The E-eigenvalues of a graph G are those of its eccentricity matrix. The E-spectrum of G is a multiset consisting of its distinct E-eigenvalues together with their multiplicities, in which the maximum modulus is called the E-spectral radius. In this paper, we order the n-vertex trees (with given diameter) regarding to their E-spectral radii. And we identify the n-vertex trees of diameter 4 having the second minimum E-spectral radius. Then we characterize the n-vertex trees having the second minimum E-spectral radius. Furthermore, the n-vertex trees with small matching number having the minimum E-spectral radii are identified. At last, for every 0≤α≤23, all graphs whose E-spectra are contained in the interval (−α,α) are characterized.