On group-valued potential differences and flows in a signed graph

A signed graph $\Sigma =(G,\sigma )$ is a graph associated with a signature $\sigma$ on each edge (positive or negative). Let $\Gamma$ be an additive abelian group and $\tau$ be an orientation of $\Sigma$. Let $f:E\to \Gamma$ be a mapping. Then $f$ is a $\Gamma$-flow if for each vertex $\sum _{(v,e)\in E\left(v\right)}\tau (v,e)f\left(e\right)=0,$ $f$ is a $\Gamma$-tension if for every signed circuit $C=v_1{e}_1{v}_2{e}_2\cdots v_k{e}_k{v}_1$, $\sum _{i=1}^k\left(-\tau (v_i,e_i)\prod _{j=1}^{i-1}\sigma \left(e_j\right)\right)f\left(e_i\right)=0,$ and $f$ is a $\Gamma$-potential difference if $f$ is a $\Gamma$-tension such that for every closed walk $W=(v_1{e}_1{v}_2{e}_2\cdots v_k{e}_k{v}_1)$ around an unbalanced circuit, $\sum _{i=1}^{k}f\left(e_i\right)\in \{2x:x\in \Gamma \}.$ We say a mapping $f:E\bigcup V\to \Gamma$ is a mixed $\Gamma$-flow if for each vertex $v$, $\sum _{(v,e)\in E\left(v\right)}\tau (v,e)f\left(e\right)=f\left(v\right).$ Denote by $F_{\Sigma }$, $P_{\Sigma }$ and $B_{\Sigma }$ the sets of $\Gamma$-flows, $\Gamma$-potential differences, and mixed $\Gamma$-flows of $\Sigma$ respectively.

In this paper, we use a new method to determine the values of $\left|F_{\Sigma }\right|$ and $\left|P_{\Sigma }\right|$. Furthermore, we show that $B_{\Sigma }/F_{\Sigma }$ is isomorphic to $P_{\Sigma }$ by a new decomposition of $\Gamma$-potential difference.