The generalized 3-connectivity of a family regular networks

The generalized \(k\)-connectivity of a graph \(G\), denoted by \(\kappa_k(G)\), is the minimum number of internally edge disjoint \(S\)-trees for any \(S\subseteq V(G)\) with \(|S|=k\). The generalized \(k\)-connectivity is a natural extension of the classical connectivity and plays a key role in applications related to the modern interconnection networks. In this paper, we firstly introduce a family of regular networks \(H_n\) that can be obtained from several subgraphs \(G_n^1, G_n^2, \cdots, G_n^{t_n}\) by adding a matching, where each subgraph \(G_n^i\) is isomorphic to a particular graph \(G_n\) (\(1\le i\le t_n\)). Then we determine the generalized 3-connectivity of \(H_n\). As applications of the main result, the generalized 3-connectivity of some two-level interconnection networks, such as the hierarchical star graph \(HS_n\), the hierarchical cubic network \(HCN_n\) and the hierarchical folded hypercube \(HFQ_n\), are determined directly.