Counting flows of b-compatible graphs

Abstract

Kochol introduced the assigning polynomial $F(G,\alpha ;k)$ to count nowhere-zero $(A,b)$-flows of a graph G, where A is a finite Abelian group and α is a $\{0,1\}$-assigning from a family $\Lambda \left(G\right)$ of certain nonempty vertex subsets of G to $\{0,1\}$. We introduce the concepts of b-compatible graph and b-compatible broken bond to give an explicit formula for the assigning polynomials and to examine their coefficients. More specifically, for a function $b:V\left(G\right)\to A$, let ${\alpha }_{G,b}$ be a $\{0,1\}$-assigning of G such that for each $X\in \Lambda \left(G\right)$, ${\alpha }_{G,b}\left(X\right)=0$ if and only if ${\sum }_{v\in X}b\left(v\right)=0$. We show that for any $\{0,1\}$-assigning α of G, if there exists a function $b:V\left(G\right)\to A$ such that G is b-compatible and $\alpha ={\alpha }_{G,b}$, then the assigning polynomial $F(G,\alpha ;k)$ has the b-compatible spanning subgraph expansion$F(G,\alpha ;k)=\sum _{\begin{array}{c}S\subseteq E\left(G\right),\\ G-S\text{ is}\text{b}\text{-compatible}\end{array}}{(-1)}^{\left|S\right|}{k}^{m(G-S)},$ and is the following form$F(G,\alpha ;k)=\sum _{i=0}^{m\left(G\right)}{(-1)}^i{a}_i(G,\alpha )k^{m\left(G\right)-i},$ where each $a_i(G,\alpha )$ is the number of subsets S of $E\left(G\right)$ having i edges such that $G-S$ is b-compatible and S contains no b-compatible broken bonds with respect to a total order on $E\left(G\right)$. Applying the counting interpretation, we also obtain unified comparison relations for the signless coefficients of assigning polynomials. Namely, for any $\{0,1\}$-assignings $\alpha ,{\alpha }^{\prime }$ of G, if there exist functions $b:V\left(G\right)\to A$ and $b^{\prime }:V\left(G\right)\to A^{\prime }$ such that G is both b-compatible and $b^{\prime }$-compatible, $\alpha ={\alpha }_{G,b}$, ${\alpha }^{\prime }={\alpha }_{G,b^{\prime }}$ and $\alpha \left(X\right)\le {\alpha }^{\prime }\left(X\right)$ for all $X\in \Lambda \left(G\right)$, then$a_i(G,\alpha )\le a_i(G,{\alpha }^{\prime })\text{ for }i=0,1,\dots ,m\left(G\right).$