Eccentricity spectral radius of t-clique trees with given diameter

Abstract

The eccentricity matrix ɛ(G) of a graph G is obtained from the distance matrix D(G) by retaining only for each row and each column the largest distance, and setting the remaining elements as 0. In this paper, we firstly show that the eccentricity matrix of clique trees is irreducible. We identify the t-clique trees with given diameter odd d having the maximum ɛ-spectral radius, and the corresponding extremal graphs are also determined. We determine the upper bounds for the ɛ-spectral radius of t-clique trees which d and n satisfy that one is odd and the other is even. Finally, we propose some potential topics for further study.