The Tenacity of Generalized Petersen Graphs

Abstract

. Communication networks can be represented as graphs, where vertices represent network nodes and edges represent connections between them. Various graph theory parameters, such as connectivity, toughness, tenacity, binding number, scattering number, and integrity, were presented to assess the vulnerability of networks. Calculating the values of these vulnerability parameters can be challenging, particularly for certain classes of graphs, such as Generalized Petersen Graphs ( G P G ), due to their diverse structures. This paper establishes upper and lower bounds for the tenacity of G P G . We demonstrate a lower bound of 1 for the tenacity ( ( ) ) , ( k n G P G T ), across all values of n and k. Additionally, we explore the tenacity values of G P G and present a general upper bound for the tenacity value in this graph type. By using the relationship between the tenacity parameter and the connectivity ( ) G κ and toughness ( ) G t parameters, we also update some theorems related to the connectivity and toughness of GPG .