A search for the minimum value of Balaban index

In this paper we consider graphs of order n with minimum Balaban index. Although the index was introduced 30 years ago, its minimum value and corresponding extremal graphs are still unknown, and it is unlikely that they can be precisely determined soon due to the mathematical intractability of the index. We show that this value is of order ? ( n - 1 ) . For small values of n we determine the extremal graphs and we observe that they are similar to dumbbell graphs. We find out that in the class of balanced dumbbell graphs those with clique sizes π / 2 4 n + o ( n ) and the path length n - o ( n ) have asymptotically the smallest value. We study dumbbell-like graphs in more detail, and we propose several conjectures regarding the structure of the extremal graphs.