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 ≤ 2 3 , all graphs whose E -spectra are contained in the interval ( − α , α ) are characterized.