Total eccentricity index of graphs with fixed number of pendant or cut vertices

TL;DR

This paper establishes sharp bounds for the total eccentricity index of connected graphs with fixed numbers of pendant or cut vertices, advancing understanding of graph eccentricity measures.

Contribution

It provides the first sharp bounds on the total eccentricity index for graphs with specified pendant or cut vertices, including new bounds for particular cases.

Findings

Sharp lower and upper bounds for graphs with fixed pendant vertices

Sharp bounds for graphs with fixed cut vertices in specific cases

Open problem proposed for bounds with intermediate cut vertices

Abstract

The total eccentricity index of a connected graph is defined as sum of the eccentricities of all its vertices. We denote the set of all connected graphs on $n$ vertices with $k$ pendant vertices by $\mathfrak{H}_{n,k}$ and denote the set of all connected graphs on $n$ vertices with $s$ cut vertices by $\mathfrak{C_{n,s}}$. In this paper, we give the sharp lower and upper bounds on the total eccentricity index over $\mathfrak{H}_{n,k}$ and the sharp lower bound for the same over $\mathfrak{C_{n,s}}$. We also provide the sharp upper bounds on the total eccentricity index over $\mathfrak{C_{n,s}}$ when $s=0,1,n-3,n-2$ and propose a problem regarding the upper bound over $\mathfrak{C_{n,s}}$ for $2\leq s\leq n-4.$