  The eccentric connectivity index $\xi^c$ is a novel distance--based molecular
structure descriptor that was recently used for mathematical modeling of
biological activities of diverse nature. It is defined as $\xi^c (G) = \sum_{v
\in V (G)} deg (v) \cdot \epsilon (v)$\,, where $deg (v)$ and $\epsilon (v)$
denote the vertex degree and eccentricity of $v$\,, respectively. We survey
some mathematical properties of this index and furthermore support the use of
eccentric connectivity index as topological structure descriptor. We present
the extremal trees and unicyclic graphs with maximum and minimum eccentric
connectivity index subject to the certain graph constraints. Sharp lower and
asymptotic upper bound for all graphs are given and various connections with
other important graph invariants are established. In addition, we present
explicit formulae for the values of eccentric connectivity index for several
families of composite graphs and designed a linear algorithm for calculating
the eccentric connectivity index of trees. Some open problems and related
indices for further study are also listed.

Combinatorics