On the distance spectral radius of unicyclic graphs with perfect matchings

Abstract

For a connected graph, the distance spectral radius is the largest eigenvalue of its distance matrix.Let U 1 2k be the graph obtained from C 3 by attaching a path of length n -3 at one vertex.Let U 2 2k be the graph obtained from C 3 by attaching a pendant edge together with k -2 paths of length 2 at the same vertex.In this paper, it is proved that U 1 2k (resp., U 2 2k ) is the unique graph with the maximum (resp., minimum) distance spectral radius among all unicyclic graphs with perfect matchings on 2k(k 5) vertices.