Minimum Eccentric Connectivity Index for Graphs with Fixed Order and Fixed Number of Pending Vertices

The eccentric connectivity index of a connected graph $G$ is the sum over all vertices $v$ of the product $d_G(v)e_G(v)$, where $d_G(v)$ is the degree of $v$ in $G$ and $e_G(v)$ is the maximum distance between $v$ and any other vertex of $G$. This index is helpful for the prediction of biological activities of diverse nature, a molecule being modeled as a graph where atoms are represented by vertices and chemical bonds by edges. We characterize those graphs which have the smallest eccentric connectivity index among all connected graphs of a given order $n$. Also, given two integers $n$ and $p$ with $p\leq n-1$, we characterize those graphs which have the smallest eccentric connectivity index among all connected graphs of order $n$ with $p$ pending vertices.