Eccentric connectivity index

TL;DR

The paper explores the eccentric connectivity index, a new molecular structure descriptor, analyzing its mathematical properties, extremal graphs, bounds, and providing algorithms and formulas for various graph families.

Contribution

It offers a comprehensive survey of the index's properties, extremal graphs, bounds, and algorithms, supporting its use in biological activity modeling.

Findings

Identified extremal trees and unicyclic graphs for the index.

Established bounds and connections with other invariants.

Developed a linear algorithm for trees.

Abstract

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.