  For a graph with edge ordering, a linear order on the edge set, we obtain a
permutation of vertices by considering the edges as transpositions of
endvertices. It is known from D\'enes' results that the permutation of a tree
is a full cyclic for any edge ordering. As a corollary, D\'enes counted up the
number of representations of a full cyclic permutation by means of product of
the minimal number of transpositions. Moreover, a graph with an edge ordering
which the permutation is a full cyclic is characterized by graph embedding.
  In this article, we consider an analogy of these results for signed graphs
and hyperoctahedral groups. We give a necessary and sufficient condition for a
signed graph to have an edge ordering such that the permutation is an even (or
odd) full cyclic. We show that the edge ordering of the signed tree with some
loops always gives an even (or odd) full cyclic permutation and count up the
number of representations of an odd full cyclic permutation by means of product
of the minimal number of transpositions.
