Symmetric Property and Reliability of Balanced Hypercube

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
$BH_n (n\ge 2)$
.