Signed Graphs and Signed Cycles of Hyperoctahedral Groups

TL;DR

This paper explores the properties of signed graphs and hyperoctahedral groups, establishing conditions for edge orderings that produce full cyclic permutations and counting minimal transposition representations.

Contribution

It extends Dénes' results to signed graphs and hyperoctahedral groups, providing necessary and sufficient conditions for full cyclic permutations and enumeration of minimal transposition factorizations.

Findings

Signed graphs can have edge orderings producing even or odd full cyclic permutations.

Signed trees with loops always yield such full cyclic permutations.

The number of minimal transposition representations of odd full cyclic permutations is counted.

Abstract

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.