On eccentric connectivity index

TL;DR

This paper explores mathematical properties and bounds of the eccentric connectivity index in graphs, providing insights into extremal trees with specific eccentric connectivity index values.

Contribution

It establishes bounds and characterizes extremal trees for the eccentric connectivity index, advancing theoretical understanding of this graph invariant.

Findings

Derived lower and upper bounds for the eccentric connectivity index.

Identified trees with extremal eccentric connectivity indices based on diameter and pendent vertices.

Characterized trees with the minimum and maximum eccentric connectivity indices.

Abstract

The eccentric connectivity index, proposed by Sharma, Goswami and Madan, has been employed successfully for the development of numerous mathematical models for the prediction of biological activities of diverse nature. We now report mathematical properties of the eccentric connectivity index. We establish various lower and upper bounds for the eccentric connectivity index in terms of other graph invariants including the number of vertices, the number of edges, the degree distance and the first Zagreb index. We determine the n-vertex trees of diameter with the minimum eccentric connectivity index, and the n-vertex trees of pendent vertices, with the maximum eccentric connectivity index. We also determine the n-vertex trees with respectively the minimum, second-minimum and third-minimum, and the maximum, second-maximum and third-maximum eccentric connectivity indices for