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

Abstract

The $PI$ index of a graph $G$ is defined by $PI\left(G\right)={\sum }_{e=(u,v)\in E}^{}[m_u\left(e\right|G)+m_v\left(e\right|G)]$ where $m_u\left(e\right|G)$ be the number of edges in $G$ lying closer to vertex $u$ than to vertex $v$ and $m_v\left(e\right|G)$ 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.