# Extremal conjugated unicyclic and bicyclic graphs with respect to total-eccentricity index

## TL;DR

This paper investigates the extremal properties of conjugated unicyclic and bicyclic graphs concerning their total-eccentricity index, extending previous work on extremal trees and conjugated trees.

## Contribution

It introduces new results characterizing extremal conjugated unicyclic and bicyclic graphs for the total-eccentricity index.

## Key findings

- Identifies extremal conjugated unicyclic graphs with maximum and minimum total-eccentricity index.
- Identifies extremal conjugated bicyclic graphs with maximum and minimum total-eccentricity index.
- Extends known extremal graph results to conjugated unicyclic and bicyclic cases.

## Abstract

Let $G$ be a molecular graph. The total-eccentricity index of graph $G$ is defined as the sum of eccentricities of all vertices of $G$. %In [R. Farooq, M.A. Malik, J. Rada, Extremal graphs with respect to total-eccentricity index, 2017, submitted], the authors studied The extremal trees, unicyclic and bicyclic graphs, and extremal conjugated trees with respect to total-eccentricity index are known. In this paper, we extend these results and study the extremal conjugated unicyclic and bicyclic graphs with respect to total-eccentricity index.