On the Wiener Index of Unicyclic Graphs with Fixed Diameter

The Wiener index is defined as W(G)=∑u,vV(G)dG(u,v), where dG(u,v) is the distance between u and v in G. In this paper, we obtain the graph with the least Wiener index among all the unicyclic graphs with n vertices and diameter d. Moreover, if 4≤d≤n－3, d≡0(mod 2), then the unicyclic graphs with the second least Wiener index are obtained.