  Motivated by the recent developments on the complexity of
non-com\-mu\-ta\-tive determinant and permanent [Chien et al.\ STOC 2011,
Bl\"aser ICALP 2013, Gentry CCC 2014] we attempt at obtaining a tight
characterization of hard instances of non-commutative permanent.
  We show that computing Cayley permanent and determinant on weight\-ed
adjacency matrices of graphs of component size six is $\#{\sf P}$ complete on
algebras that contain $2\times 2$ matrices and the permutation group $S_3$.
Also, we prove a lower bound of $2^{\Omega(n)}$ on the size of branching
programs computing the Cayley permanent on adjacency matrices of graphs with
component size bounded by two. Further, we observe that the lower bound holds
for almost all graphs of component size two.
  On the positive side, we show that the Cayley permanent on graphs of
component size $c$ can be computed in time $n^{c{\sf poly}(t)}$, where $t$ is a
parameter depending on the labels of the vertices.
  Finally, we exhibit polynomials that are equivalent to the Cayley permanent
polynomial but are easy to compute over commutative domains.
