Extremal graphs with respect to the total-eccentricity index

TL;DR

This paper investigates extremal graphs, including trees and certain cyclic graphs, to identify those with maximum or minimum total-eccentricity index, advancing understanding of graph eccentricity properties.

Contribution

It characterizes extremal trees, unicyclic, bicyclic, and conjugated trees concerning the total-eccentricity index, providing new insights into graph eccentricity extremal structures.

Findings

Identifies extremal trees with respect to total-eccentricity index.

Finds extremal unicyclic and bicyclic graphs for the same index.

Determines extremal conjugated trees based on total-eccentricity index.

Abstract

In a connected graph G, the distance between two vertices of G is the length of a shortest path between these vertices. The eccentricity of a vertex u in G is the largest distance between u and any other vertex of G. The total-eccentricity index {\tau}(G) is the sum of eccentricities of all vertices of G. In this paper, we find extremal trees, unicyclic and bicyclic graphs with respect to total-eccentricity index. Moreover, we find extremal conjugated trees with respect to total-eccentricity index.