Generalized 4-connectivity of alternating group networks

Connectivity is a fundamental attribute crucial for the efficiency of interconnection networks, especially in domains requiring robust communication infrastructures. A natural generalization of the connectivity is the generalized connectivity introduced by Hager (J Combin Theory Ser B 38:179–189, 1985). This paper explores the problem of determining the generalized 4-connectivity of the alternating group network ( \(AN_n\) $A{N}_n$ ), motivated by the challenges inherent in designing resilient and efficient networks. We prove that for any set of four vertices in \(AN_n\) $A{N}_n$ , there exist \(n-2\) $n-2$ trees in \(AN_n\) $A{N}_n$ having in common exactly these four vertices, offering insights into the network’s structural characteristics with implications for applications demanding resilient communication paths. Additionally, we establish the value of the generalized 4-edge-connectivity of \(AN_n\) $A{N}_n$ .