Component edge connectivity and extra edge connectivity of alternating group networks

Abstract

The l-component edge connectivity of a graph G, denoted by ${c\lambda _l (G)}$, is the minimum number of edges whose removal from G results in a disconnected graph with at least l-components. The h-extra edge connectivity of a graph G, denoted by $\lambda _h(G)$, is the minimum number of edges whose removal from G results in a disconnected graph and each component has at least $h+1$ vertices. In this paper, we determine the l-component edge connectivity and the h-extra edge connectivity of alternating group networks for some small values. For l-component edge connectivity, we prove that $c\lambda _3(AN_n)=2n-3$ for $n\ge 3$, $c\lambda _4(AN_n)=3n-6$ for $n\ge 4$, and $c\lambda _5(AN_n)=4n-8$ for $n\ge 4$. For h-extra edge connectivity, we prove that $\lambda _1(AN_n)=2n-4$, $\lambda _2(AN_n)=3n-9$ and $\lambda _3(AN_n)=4n-12$ for $n\ge 6$.