Cacti with extremal PI Index

The vertex PI index PI(G) = \sum_{xy \in E(G)} [n_{xy}(x)‎ + ‎n_{xy}(y)] is a distance-based molecular structure descriptor‎, ‎where n_{xy}(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 ۱۹۴۷‎. ‎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.

Keywords:

‎Distance‎ , ‎Extremal bounds‎ , ‎PI index‎ , ‎Cacti