Signed Graphs and Signed Cycles of Hyperoctahedral Groups

Ryo Uchiumi

Published 2022-11-10Version 1

J. D\'enes gave a bijection between the set of edge-labeled trees on $\{1,\ldots,n\}$ and the set of sequences consisting of $n-1$ transpositions such that the product is an $n$-cycle. As a corollary, D\'enes proved that the number of trees on $\{1,\ldots,n\}$ is equal to the number of representations of an $n$-cycle by means of product of $n-1$ transpositions. In this article, we consider an analogy of D\'enes' results for signed trees and hyperoctahedral groups.