On the largest and least eigenvalues of eccentricity matrix of trees

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