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), motivated by the challenges inherent in designing resilient and efficient networks. We prove that for any set of four vertices in AN_n, there exist n-2 trees in AN_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.