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

TL;DR

This paper analyzes the cyclic connectivity of Cayley graphs generated by unicyclic triangle-free graphs, providing an exact formula for their cyclic connectivity, which enhances understanding of their fault tolerance in network applications.

Contribution

It introduces the cyclic connectivity of Cayley graphs generated by unicyclic triangle-free graphs and derives an exact formula for this property.

Findings

Cyclic connectivity of UG_n is 4n-8 for n ≥ 4.

Provides a precise measure of fault tolerance for these Cayley graphs.

Enhances understanding of network robustness based on cyclic connectivity.

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 $\left\{ 1,2,\dots,n\right\}$ and a set $\mathcal{T}$ of transpositions of $Sym(n)$. Let $G(\mathcal{T})$ be the graph on vertex set $\left\{ 1,2,\dots,n\right\}$ and edge set $\left\{ij\colon(ij)\in \mathcal{T}\right\}$. 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 $\kappa_{c}(UG_{n})=4n-8$ for $n\ge 4 $.