Extreme tenacity of graphs with given order and size

TL;DR

This paper investigates the extremal values of graph tenacity for graphs with fixed order and size, providing solutions for minimum and maximum tenacity, especially in trees and unicyclic graphs, to aid in designing stable networks.

Contribution

It offers a complete solution for the minimum tenacity problem and characterizes the maximum tenacity in trees and unicyclic graphs, advancing understanding of network stability measures.

Findings

Complete solution for minimum tenacity of graphs with given order and size

Determination of maximum tenacity in trees and unicyclic graphs

Identification of extremal graphs for stability and cost-effective network design

Abstract

Computer or communication networks are so designed that they do not easily get disrupted under external attack and, moreover, these are easily reconstructible if they do get disrupted. These desirable properties of networks can be measured by various graph parameters, such as connectivity, toughness, scattering number, integrity, tenacity, rupture degree and edge-analogues of some of them. Among these parameters, the tenacity and rupture degree are two better ones to measure the stability of a network. In this paper we consider two extremal problems on the tenacity of graphs: Determine the minimum and maximum tenacity of graphs with given order and size. We give a complete solution to the first problem, while for the second one, it turns out that the problem is much more complicated than that of the minimum case. We determine the maximum tenacity of trees and unicyclic graphs with given order and show the corresponding extremal graphs. These results are helpful in constructing stable networks with lower costs. The paper concludes with a discussion of a related problem on the edge vulnerability parameters of graphs.