Conformal Decomposition of Integral Tensions and Potentials of Signed Graphs
Beifang Chen · SIAM Journal on Discrete Mathematics · 2017
Let $\Gamma$ be a subgroup of an integral chain group on a set $E$. A nonzero chain $g$ of $\Gamma$ is said to be conformally decomposable if there exist nonzero chains $g_1,g_2$ of $\Gamma$ such that $g=g_1+g_2$ and $g_1(e)g_2(e)\geq 0$ for all $e\in E$. For a signed graph $\Sigma$ with edge set $E$, there are two subgroups $F(\Sigma,{\Bbb Z})$ and $T(\Sigma,{\Bbb Z})$ of the 1-chain group $C_1(\Sigma,{\Bbb Z})$, known as the flow lattice and tension lattice of $\Sigma$. The conformally indecomposable flows of $F(\Sigma,{\Bbb Z})$ are classified in [B. Chen and J. Wang, https://arXiv.org/abs/1112.0642, 2011; B. Chen, J. Wang, and T. Zaslavsky, Discrete Math., 340 (2017), pp. 1271--1786] as signed-graphic circuit flows and a class of characteristic vectors of certain directed Eulerian cycle-trees. In this paper we classify conformally indecomposable tensions of $T(\Sigma,{\Bbb Z})$ as characteristic vectors of signed-graphic directed bonds and a class of characteristic vectors of directed semi-bonds and directed hyper-bonds. The half-spin structures ($\pm\frac{1}{2}$-potential functions) of $\Sigma$ correspond to characteristic vectors of directed hyper-bonds. A byproduct is the classification of conformally indecomposable integral potentials.