علیرضا عطایی

Let $G$ be a simple connected graph with vertex set $V (G)$. The eccentric connectivity index of $G$, denoted by $\xi^c(G)$, is defined as $\sum_{v\in V (G)} \deg_G(v)\varepsilon_G(v)$, where $\deg_G(v)$ is the degree of the vertex $v$ and $\varepsilon_G(v)$ is defined as the maximum distance from $v$ to any other vertex of $G$. In this thesis, Sharp lower and asymptotic upper bound for all graphs are given and various connections with other important graph invariants are established. we present the extremal trees with maximum and minimum eccentric connectivity index subject to the certain graph constraints. In addition, we present explicit formulae for the values of eccentric connectivity index for several families of composite graphs.