Wiener Index of Graphs and Their Line Graphs

Abstract

The Wiener index W(G) of a graph G is a distance-based topological index defined as the sum of distances between all pairs of vertices in G. It is shown that for λ=2 there is an infinite family of planar bipartite chemical graphs G of girth 4 with the cyclomatic number λ, but their line graphs are not chemical graphs, and for λ≥2 there are two infinite families of planar nonbipartite graphs G of girth 3 with the cyclomatic number λ; the three classes of graphs have the property W(G)=W(L(G)), where L(G) is the line graph of G. © 2013 Operations Research Society of China, Periodicals Agency of Shanghai University, and Springer-Verlag Berlin Heidelberg.