cacti with extremal pi index

The vertex pi index pi(g)=∑xy∈e(g)[nxy(x)‎+‎nxy(y)] is a distance-based molecular structure descriptor‎, ‎where nxy(x) denotes the number of vertices which are closer to the vertex x than to the vertex y and which has been the considerable research in computational chemistry dating back to harold wiener in 1947‎. ‎a connected graph is a cactus if any two of its cycles have at most one common vertex‎. ‎in this paper‎, ‎we completely determine the extremal graphs with the greatest and smallest vertex pi indices mong all cacti with a fixed number of vertices‎. ‎as a consequence‎, ‎we obtain the sharp bounds with corresponding extremal cacti and extend a known result‎.