Symmetric Property and Reliability of Balanced Hypercube

Abstract

Huang and Wu in [IEEE Transactions on Computers 46 (1997) 484-490] introduced the balanced hypercube $BH_n$ as an interconnection network topology for computing systems, and they proved that $BH_n$ is vertex-transitive. However, some other symmetric properties, say edge-transitivity and arc-transitivity, of $BH_n$ remained unknown. In this paper, we solve this problem and prove that $BH_n$ is an arc-transitive Cayley graph. Using this, we also investigate some reliability measures, including super-connectivity, cyclic connectivity, etc., in $BH_n$ . First, we prove that every minimum edge-cut of $BH_n (n\ge 2)$ isolates a vertex, and every minimum vertex-cut of $BH_n (n\ge 3)$ isolates a vertex. This is stronger than that obtained by Wu and Huang which shows the connectivity and edge-connectivity of $BH_n$ are $2n$ . Second, Yang [Applied Mathematics and Computation 219 (2012) 970-975.] proved that for $n\ge 2$ , the super-connectivity of $BH_n$ is $4n-4$ and the super edge-connectivity of $BH_n$ is $4n-2$ . In this paper, we proved that $BH_n (n\ge 2)$ is super- $\lambda^{\prime }$ but not super- $\kappa^{\prime }$ . That is, every minimum super edge-cut of $BH_n (n\ge 2)$ isolates an edge, but the minimum super vertex-cut of $BH_n (n\ge 2)$ does not isolate an edge. Third, we also obtain that for $n\ge 2$ , the cyclic connectivity of $BH_n$ is $4n-4$ and the cyclic edge-connectivity of $BH_n$ is $4(2n-2)$ . That is, to become a disconnected graph which has at least two components containing cycles, we need to remove at least $4n-4$ vertices (resp. $4(4n-2)$ edges) from $BH_n (n\ge 2)$ .