On the largest and least eigenvalues of eccentricity matrix of trees

Abstract

The eccentricity matrix ε(G) of a graph G is constructed from the distance matrix of G by keeping only the largest distances for each row and each column. This matrix can be interpreted as the opposite of the adjacency matrix obtained from the distance matrix by keeping only the distances equal to 1 for each row and each column. The ε-eigenvalues of a graph G are those of its eccentricity matrix ε(G). Wang et al. [24] proposed the problem of determining the maximum ε-spectral radius of trees with given order. In this paper, we consider the above problem of n-vertex trees with given diameter. The maximum ε-spectral radius of n-vertex trees with fixed odd diameter is obtained, and the corresponding extremal trees are also determined. Recently, Wei et al. [22] determined all connected graphs on n vertices of maximum degree less than n−1, whose least eccentricity eigenvalues are in [−22,−2]. Denote by Sn the star on n vertices. For tree T with order n≥3, it [22] was proved that εn(T)≤−2 with equality if and only if T≅Sn. According to the above results, the trees of order n≥3 with least ε-eigenvalues in [−22,0) are only Sn. Motivated by [22], we determine the trees with least ε-eigenvalues in [−2−13,−22).