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

TL;DR

This work investigates the cyclic connectivity of Cayley graphs generated by unicyclic triangle-free transposition sets, denoted $UG_n$. By exploiting the hierarchical structure of $UG_n$ and related families such as $MB_n$, the authors establish tight bounds and prove the exact value $κ_c(UG_n)=4n-8$ for all $n≥4$, with separate treatments for small cases ($n=4,5$). The key link is showing that $κ_c(UG_n)$ matches the 2-good-neighbor connectivity, $κ^2(UG_n)=4n-8$, aided by the girth $g(UG_n)=4$, thereby confirming high fault tolerance in these networks. Overall, the results provide a precise reliability metric for Cayley-graph-based interconnection networks and highlight the structural role of unicyclic triangle-free generating graphs in determining 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 $κ_{c}(UG_{n})=4n-8$ for $n\ge 4 $.

Figures (5)

• Figure 1: The modified bubble-sort graph $MB_{4}$
• Figure 2: The structure of $A$.
• Figure 3: The explanation of $G_{a}$,$G_{b}$ and $G_{c}$.
• Figure 4: The boundary of inequality is a tight statement
• Figure 5: The collection ${ MB_{4}^{i} - F_{i} : i \in [3] }$ of all possible structures.

Theorems & Definitions (16)

• Lemma 2.1
• Lemma 2.2
• Lemma 2.3
• Lemma 2.4
• Lemma 2.5
• Lemma 2.6
• Lemma 2.7
• Lemma 2.8
• Lemma 2.9
• proof