The Wiener Indices Of Unicyclic Graphs

Let G = (V,E) be a simple connected graph with vertex set V and edge set E. The Wiener index W(G) of G is the sum of distances between all pairs of vertices in G, i.e.,where d_G(u,v) is the distance between vertices u and v in G. Wiener index is one of the classic topological indices (graph invariants) in chemical graph theory. It has been successfully used in theoretical chemistry for quantitative structure-property relations (QSPR) and quantitative structure-property relations (QSAR), and also in studying communication networks.In this paper, we study the Wiener indices of unicyclic graphs. Fristly, we give a formulation for calculating the Wiener index of an unicyclic graph according its struction. And then, using this formulation, we characterize the graphs with the largest, the smallest, the second largest , the second smallest, the third largest and the third smallest Wiener indices among all the unicyclic graphs of order n.