Tightness of Harary Graphs

Abolfazl Javan, Ali Moeini () and Mohammad Shekaramiz
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Abolfazl Javan: Department of Algorithms and Computation, School of Engineering Sciences, College of Engineering, University of Tehran, 16 Azar, Tehran 1417614411, Iran
Ali Moeini: Department of Algorithms and Computation, School of Engineering Sciences, College of Engineering, University of Tehran, 16 Azar, Tehran 1417614411, Iran
Mohammad Shekaramiz: Machine Learning and Drone Laboratory, Electrical and Computer Engineering Department, Utah Valley University, Orem, UT 84058, USA

Mathematics, 2024, vol. 12, issue 18, 1-12

Abstract: In the design of real-world networks, researchers evaluate various structural parameters to assess vulnerability, including connectivity, toughness, and tenacity. Recently, the tightness metric has emerged as a potentially superior vulnerability measure, although many related theorems remain unknown due to its novelty. Harary graphs, known for their maximum connectivity, are an important class of graph models for network design. Prior work has evaluated the vulnerability of three types of Harary graphs using different parameters, but the tightness metric has not been thoroughly explored. This article aims to calculate the tightness values for all three types of Harary graphs. First, it will attempt to calculate the lower bound for the value of the tightness parameter in Harary graphs using existing lemmas and theorems. Then, by presenting new lemmas and theorems, we will try to find the exact value or upper bound for this parameter in Harary graphs. For the first type of Harary graph, the tightness is precisely determined, while for the second and third types, upper bounds are provided due to structural complexity. The lemmas, theorems, and proof methods presented in this research may be used to calculate other graph and network parameters. However, the newness of the tightness parameter means that further research is needed to fully characterize its properties.

Keywords: network vulnerability; tightness; Harary graphs; connectivity; Hamiltonian circuit (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2024
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