Polynomials Counting Nowhere-Zero Chains in Graphs

Abstract

We introduce polynomials counting nowhere-zero chains in graphs – nonhomogeneous analogues of nowhere-zero flows. For a graph $G$, an Abelian group $A$, and $b:V(G)\to A$, let $\alpha_{G,b}$ be a mapping from $\Lambda(G)$ (a family of vertex sets of connected subgraphs of $G$ satisfying an additional condition) to $\{0,1\}$ such that for each $X\in\Lambda(G)$, $\alpha_{G,b}(X)=0$ if and only if $\sum_{v\in X}b(v)=0$. We prove that there exists a polynomial function $F(G,\alpha;k)$ ($\alpha=\alpha_{G,b}$) of $k$ such that for any Abelian group $A'$ of order $k$ and each $b':V(G)\to A'$ satisfying $\alpha_{G,b'}=\alpha$, $F(G,\alpha;k)$ equals the number of nowhere-zero $A'$-chains $\varphi$ in $G$ having boundaries equal to $b'$. In particular $F(G,\alpha;k)$ is the flow polynomial of $G$ if $\alpha(X)=0$ for each $X\in\Lambda(G)$. Finally we characterize $\alpha$ for which $F(G,\alpha;k)$ is nonzero and show that in this case $F(G,\alpha;k)$ has the same degree as the flow polynomial of $G$.