Vulnerability Measures and Zagreb Indices of Graphs

TL;DR

This paper derives bounds for vulnerability measures like closeness in graphs, linking them with Zagreb indices, and characterizes extremal graphs for various classes such as triangle-free and trees.

Contribution

It introduces new bounds for vulnerability measures based on Zagreb indices and identifies graphs that attain these bounds across different graph classes.

Findings

Sharp bounds for closeness and generalized closeness vulnerability measures.

Formulas for these measures in triangle- and quadrangle-free graphs with diameter ≤ 3.

Bounds for trees and graphs with girth ≥ 7, attained by graphs with diameter ≤ 4.

Abstract

This paper establishes sharp bounds for the vulnerability measures of closeness and generalized closeness in graphs and identifies graphs that attain these bounds. It further develops bounds incorporating Zagreb indices for triangle- and quadrangle-free graphs, yielding formulas for closeness and generalized closeness in such graphs with diameter at most 3. Moreover, using Zagreb indices, we derive bounds for trees and connected graphs with girth at least 7, which are attained by graphs with diameter at most 4. Finally, formulas for closeness and generalized closeness in specific trees are established using Zagreb indices.