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 R C π G (x) = ∑ y ∈ V G d (y) H y x , where d (y) denotes the degree of y and H y x denotes the hitting time for random walk starting at y to visit x. In this paper, we first obtain the sharp lower bound of R C π G (x) among all trees with order n and given diameter, and then determine the sharp upper bound of R C π G (x) among all unicyclic graphs with order n and given diameter. Furthermore, all the corresponding extremal graphs are identified. • Present the sharp lower bound of weighted reverse cover cost among all trees of order n and diameter d. • Present the sharp upper bound of weighted reverse cover cost among all unicyclic graphs of order n and diameter d. • Characterize the corresponding extremal graphs. [ABSTRACT FROM AUTHOR]