Balaban Index of Cubic Graphs

Balaban index is defined as J(G) = m/m-n+2 Sigma 1/root w(u).w(v), where the sum is taken over all edges of a connected graph G, n and m are the cardinalities of the vertex and the edge set of G, respectively, and w(u) (resp. w(v)) denotes the sum of distances from u (resp. v) to all the other vertices of G. In this paper, we give an upper bound for the Balaban index of r-regular graphs on n vertices. Then we concentrate on cubic graphs. We give a better upper bound for fullerene graphs and we show that the Balaban index tends to zero as the number of vertices increases. This means that Balaban index does not distinguish well the fullerene graphs when they are sufficiently large. We conclude the paper with a conjecture on the lower bound of Balaban index for cubic graphs.