Solutions for two conjectures on the eigenvalues of the eccentricity matrix, and beyond

The eccentricity matrix e ( 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. In this paper we focus on the eccentricity matrix of graphs. Let T be an n -vertex tree and let e n ( T ) be the least e -eigenvalue of T . On the one hand, we determine the n -vertex trees with the minimum e -spectral radius. On the other hand, for n ⩾ 3 , we show that e n ( T ) ⩽ − 2 with equality if and only if T is a star. As a consequence, we solve two conjectures proposed by Wang et al. (2018). Furthermore, we identify all the trees with given order and diameter having the minimum e -spectral radius . Finally, we determine all the n -vertex connected graphs whose maximum degrees are less than n − 1 and least e -eigenvalues are in [ − 2 2 , − 2 ] .