Maximum Eccentric Connectivity Index for Graphs with Given Diameter

TL;DR

This paper characterizes the connected graphs with the maximum eccentric connectivity index for given order and diameter, providing a complete description of extremal structures in this graph invariant.

Contribution

It offers a complete characterization of graphs with the maximum eccentric connectivity index for specified order and diameter, filling a gap in extremal graph theory.

Findings

Identifies extremal graphs with maximum eccentric connectivity index for fixed diameter.

Provides a characterization of extremal graphs for given order without diameter constraint.

Establishes the maximum eccentric connectivity index for graphs of given order and diameter.

Abstract

The eccentricity of a vertex $v$ in a graph $G$ is the maximum distance between $v$ and any other vertex of $G$. The diameter of a graph $G$ is the maximum eccentricity of a vertex in $G$. The eccentric connectivity index of a connected graph is the sum over all vertices of the product between eccentricity and degree. Given two integers $n$ and $D$ with $D\leq n-1$, we characterize those graphs which have the largest eccentric connectivity index among all connected graphs of order $n$ and diameter $D$. As a corollary, we also characterize those graphs which have the largest eccentric connectivity index among all connected graphs of a given order $n$.