Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: The Dimension of the Negative Cycle Vectors of Signed Graphs   A "signed graph" is a graph $\Gamma$ where the edges are assigned sign
labels, either "$+$" or "$-$". The sign of a cycle is the product of the signs
of its edges. Let $\mathrm{SpecC}(\Gamma)$ denote the list of lengths of cycles
in $\Gamma$. We equip each signed graph with a vector whose entries are the
numbers of negative $k$-cycles for $k\in\mathrm{SpecC}(\Gamma)$. These vectors
generate a subspace of $\mathbb R^{\mathrm{SpecC}(\Gamma)}$. Using matchings
with a strong permutability property, we provide lower bounds on the dimension
of this space; in particular, we show for complete graphs, complete bipartite
graphs, and a few other graphs that this space is all of $\mathbb
R^{\mathrm{SpecC}(\Gamma)}$.

28
The categories of this paper are: alg-geom
22
The categories of this paper are: chao-dyn
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The categories of this paper are: chem-ph
16
The categories of this paper are: stat
14
The categories of this paper are: math-ph
25
The categories of this paper are: patt-sol
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The categories of this paper are: adap-org
6
The categories of this paper are: astro-ph
3
The categories of this paper are: physics
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The categories of this paper are: ao-sci
1
The categories of this paper are: cs
11
The categories of this paper are: hep-lat
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The categories of this paper are: mtrl-th
10
The categories of this paper are: q-bio
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The categories of this paper are: plasm-ph
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The categories of this paper are: eess