Disproving a Conjecture on PI-Index of Graphs

The PI-index of a graph G is defined by PI(G) = Sigma(e=uv is an element of E)(m(u)(e) + m(v)(e)), where m(u)(e) is the number of edges in G lying closer to the vertex u than to the vertex v. In this paper, using probabilistic method we disprove a conjecture about the PI-index posed by Khalifeh et al. [Order of Magnitude of the PI index, MATCH Commun. Math. Comput. Chem. 65 (2011) 51-56].