  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.
