Bounds on the PI index of unicyclic and bicyclic graphs with given girth

The PI index of a graph G is defined by PI(G)=∑e=(u,v)∈E[mu(e|G)+mv(e|G)] where mu(e|G) be the number of edges in G lying closer to vertex u than to vertex v and mv(e|G) be the number of edges in G lying closer to vertex v than to vertex u. In this paper, we give the upper and lower bounds on the PI index of connected unicyclic and bicyclic graphs with given girth and completely characterize the corresponding extremal graphs. From our results, it is easy to get the bounds and extremal graphs of the unicyclic and bicyclic graphs.