The Maximum Hyper-Wiener Index Of Unicyclic Graphs

LetG be a simple connected graph with vertex set V(G).For two arbitrary vertices u and v of G,let dG(u,v)denote the distance between u and v,which is the number of edges on a shortest path connecting these vertices in G.Then the Wiener index W(G)and hyper-Wiener index WW(G)of G are defined as followThe Wiener index and hyper-Wiener index are two descriptors closely related to molecular branching.In this article we first give some new graphic transformations,then with these transformations identify the unicyclic graphs having the third up to seven largest hyper-Wiener indices among all unicyclic graphs with a fixed order,finally determine the graphs having the largest hyper-Wiener index among all unicyclic graphs given maximum degree.