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 arctransitivity, 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 ðn ! 2Þ isolates a vertex, and every minimum vertex-cut of BH n ðn ! 3Þ 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 ! 2, the super-connectivity of BH n is 4n À 4 and the super edge-connectivity of BH n is 4n À 2. In this paper, we proved that BH n ðn ! 2Þ is super-0 but not super-k 0 . That is, every minimum super edge-cut of BH n ðn ! 2Þ isolates an edge, but the minimum super vertex-cut of BH n ðn ! 2Þ does not isolate an edge. Third, we also obtain that for n ! 2, the cyclic connectivity of BH n is 4n À 4 and the cyclic edge-connectivity of BH n is 4ð2n À 2Þ. That is, to become a disconnected graph which has at least two components containing cycles, we need to remove at least 4n À 4 vertices (resp. 4ð4n À 2Þ edges) from BH n ðn ! 2Þ.

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Huang and Wu in [IEEE Transactions on Computers 46 (1997), pp. 484-490] introduced the balanced hypercube BH n as an interconnection network topology for computing systems. In this paper, we completely determine the full automorphism group of the balanced hypercube. Applying this, we first show that the n-dimensional balanced hypercube BH n is arc-transitive but not 2-arc-transitive whenever n ≥ 2. Then, we show that BH n is a lexicographic product of an n-valent graph X n and the null graph with two vertices, where X n is a Z n−1 2 -regular cover of the n-dimensional hypercube Q n .