Reliability analysis of godan graphs in terms of generalized 4-connectivity

Abstract

Let $G$ be a connected graph and $S\subseteq V\left(G\right)$. Denote by ${\kappa }_G\left(S\right)$ the maximum number $r$ of internally disjoint $S$-trees $T_1$, $T_2$, $\dots$, $T_r$ in $G$ such that $V\left(T_i\right)\cap V\left(T_j\right)=S$ and $E\left(T_i\right)\cap E\left(T_j\right)=0̸$ for any integers $1\le i<j\le r$. For an integer $k$ with $2\le k\le \left|V\left(G\right)\right|$, the generalized $k$-connectivity of a graph $G$, denoted by ${\kappa }_k\left(G\right)$, is defined as ${\kappa }_k\left(G\right)=min\text{{}{\kappa }_G\left(S\right)|S\subseteq V\left(G\right)$ and $\left|S\right|=k\text{}}$. The generalized $k$-connectivity of a graph is a natural extension of the classical connectivity and plays a key role in measuring the reliability of modern interconnection networks. The godan graph $E{A}_n$ is a kind of Cayley graph which possess many desirable properties. In this paper, we study the generalized 4-connectivity of $E{A}_n$ and show that ${\kappa }_4\left(E{A}_n\right)=n-1$, that is, there are $n-1$ internally disjoint $S$-trees connecting any four vertices $x,y,z$ and $w$ in $E{A}_n$, where $n\ge 3$ and $S=\{x,y,z,w\}$.