The core index of a graph

For a graph $G,$ we denote the number of connected subgraphs of $G$ by $F(G)$. For a tree $T$, $F(T)$ has been studied extensively and it has been observed that $F(T)$ has a reverse correlation with Wiener index of $T$. Based on that, we call $F(G),$ the core index of $G$. In this paper, we characterize the graphs which extremize the core index among all graphs on $n$ vertices with $k\geq 0$ connected components. We extend our study of core index to unicyclic graphs and connected graphs with fixed number of pendant vertices. We obtained the unicyclic graphs which extremize the core index over all unicyclic graphs on $n$ vertices. The graphs which extremize the core index among all unicyclic graphs with fixed girth are also obtained. Among all connected graphs on $n$ vertices with fixed number of pendant vertices, the graph which minimizes and the graph which maximizes the core index are characterized.