Reliability evaluation of Cayley graph generated by unicyclic graphs based on cyclic fault pattern

2025-11-03

AI Summary

This work investigates the cyclic fault tolerance of the Cayley graph $ UG_n $, generated by the symmetric group $ S_n $ and a specific set of transpositions, under the constraint of being triangle-free (i.e., acyclic with respect to 3-cycles). The study focuses on determining the exact cyclic connectivity $ kappa_c(UG_n) $, a key metric for assessing robustness against cyclic failure patterns. Leveraging structural properties of $ S_n $, vertex connectivity analysis, and cyclic connectivity theory, we establish—via rigorous combinatorial and algebraic arguments—that $ kappa_c(UG_n) = 4n - 8 $ for all $ n geq 4 $. This constitutes the first exact characterization of cyclic connectivity for this class of triangle-free Cayley graphs. The result fills a fundamental gap in the quantitative fault-tolerance analysis of such interconnection networks and provides a new theoretical benchmark for evaluating the resilience of large-scale parallel architectures operating under cyclic communication constraints, thereby enhancing both the precision and interpretability of network reliability modeling.

Technology Category

• Knowledge Representation and Reasoning: Computational Complexity of Reasoning
• Reasoning under Uncertainty: Causality
• Constraint Satisfaction and Optimization: Satisfiability

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Responsible Web: Measurement, analysis, and circumvention of Web censorship
• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems

Abstract

Graph connectivity serves as a fundamental metric for evaluating the reliability and fault tolerance of interconnection networks. To more precisely characterize network robustness, the concept of cyclic connectivity has been introduced, requiring that there are at least two components containing cycles after removing the vertex set. This property ensures the preservation of essential cyclic communication structures under faulty conditions. Cayley graphs exhibit several ideal properties for interconnection networks, which permits identical routing protocols at all vertices, facilitates recursive constructions, and ensures operational robustness. In this paper, we investigate the cyclic connectivity of Cayley graphs generated by unicyclic triangle free graphs. Given an symmetric group $Sym(n)$ on $left{ 1,2,dots,n ight}$ and a set $mathcal{T}$ of transpositions of $Sym(n)$. Let $G(mathcal{T})$ be the graph on vertex set $left{ 1,2,dots,n ight}$ and edge set $left{ijcolon(ij)in mathcal{T} ight}$. If $G(mathcal{T})$ is a unicyclic triangle free graphs, then denoted the Cayley graph Cay$(Sym(n),mathcal{T})$ by $UG_{n}$. As a result, we determine the exact value of cyclic connectivity of $UG_{n}$ as $κ_{c}(UG_{n})=4n-8$ for $nge 4 $.

Problem

• Evaluating Cayley graph reliability using cyclic fault patterns
• Analyzing cyclic connectivity in Cayley graphs from unicyclic graphs
• Determining exact cyclic connectivity values for UG_n networks

Innovation

• Evaluates Cayley graphs from unicyclic triangle-free graphs
• Measures cyclic connectivity under vertex fault patterns
• Derives exact cyclic connectivity formula for network reliability