On the weighted reverse cover cost of trees and unicyclic graphs with given diameter

For a vertex x of a connected graph G = (V-G, E-G), the weighted reverse cover cost of x is proposed as RC pi G(x) = Sigma(y is an element of VG) d(y)H-yx, where d(y) denotes the degree of y and H-yx denotes the hitting time for random walk starting at y to visit x. In this paper, we first obtain the sharp lower bound of RC pi G(x) among all trees with order.. and given diameter, and then determine the sharp upper bound of RC pi G(x) among all unicyclic graphs with order n and given diameter. Furthermore, all the corresponding extremal graphs are identified.