Abstract. Balaban index of a graph G is

J(G)=m(m-n+2)-1Σ(w(u).w(v))-1,

where the sum is taken over all edges uv of G, m and n are the cardinalities of the vertex and edge sets of G, respectively, and w(x) is the sum of distances from x to all the other vertices of G. In this paper we give an upper bound for J(G) when G is an r-regular graph. This bound is of type f(r)/log(n), which means that when n increases, the Balaban index of r-regular graph tends to 0. Then we show that for fullerene graphs the upper bound for Balaban index is 25.n-0.5. Finally, we show a class of cubic (=3-regular) graphs for which J(G)≤32/n. Then we construct another class of cubic graphs and we conjecture that it has the smallest Balaban index in the class of cubic graphs.