Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Flow polynomials of a signed graph   In contrast to ordinary graphs, the number of the nowhere-zero group-flows in
a signed graph may vary with different groups, even if the groups have the same
order. In fact, for a signed graph $G$ and non-negative integer $d$, it was
shown that there exists a polynomial $F_d(G,x)$ such that the number of the
nowhere-zero $\Gamma$-flows in $G$ equals $F_d(G,x)$ evaluated at $k$ for every
Abelian group $\Gamma$ of order $k$ with $\epsilon(\Gamma)=d$, where
$\epsilon(\Gamma)$ is the largest integer $d$ for which $\Gamma$ has a subgroup
isomorphic to $\mathbb{Z}^d_2$. We focus on the combinatorial structure of
$\Gamma$-flows in a signed graph and the coefficients in $F_d(G,x)$. We first
define the fundamental directed circuits for a signed graph $G$ and show that
all $\Gamma$-flows (not necessarily nowhere-zero) in $G$ can be generated by
these circuits. It turns out that all $\Gamma$-flows in $G$ can be evenly
classified into $2^{\epsilon(\Gamma)}$-classes specified by the elements of
order 2 in $\Gamma$, each class of which consists of the same number of flows
depending only on the order of the group. This gives an explanation for why the
number of $\Gamma$-flows in a signed graph varies with different
$\epsilon(\Gamma)$, and also gives an answer to a problem posed by Beck and
Zaslavsky. Secondly, using an extension of Whitney's broken circuit theory we
give a combinatorial interpretation of the coefficients in $F_d(G,x)$ for
$d=0$, in terms of the broken bonds. As an example, we give an analytic
expression of $F_0(G,x)$ for a class of the signed graphs that contain no
balanced circuit. Finally, we show that the sets of edges in a signed graph
that contain no broken bond form a homogeneous simplicial complex.

18
The categories of this paper are: econ
31
The categories of this paper are: ao-sci
29
The categories of this paper are: funct-an
17
The categories of this paper are: q-fin
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The categories of this paper are: dg-ga
11
The categories of this paper are: hep-lat
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The categories of this paper are: nucl-th
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The categories of this paper are: q-bio
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The categories of this paper are: mtrl-th
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The categories of this paper are: chao-dyn
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The categories of this paper are: stat
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The categories of this paper are: quant-ph
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The categories of this paper are: comp-gas
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The categories of this paper are: hep-ex
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The categories of this paper are: atom-ph
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The categories of this paper are: bayes-an