The Dimension of the Negative Cycle Vectors of Signed Graphs [article]

Alex Schaefer, Thomas Zaslavsky
2017 arXiv   pre-print
A "signed graph" is a graph Γ where the edges are assigned sign labels, either "+" or "-". The sign of a cycle is the product of the signs of its edges. Let SpecC(Γ) denote the list of lengths of cycles in Γ. We equip each signed graph with a vector whose entries are the numbers of negative k-cycles for k∈SpecC(Γ). These vectors generate a subspace of R^SpecC(Γ). Using matchings with a strong permutability property, we provide lower bounds on the dimension of this space; in particular, we show
more » ... or complete graphs, complete bipartite graphs, and a few other graphs that this space is all of R^SpecC(Γ).
arXiv:1706.09041v1 fatcat:iwmaippnxbdx7myazpnhhy7czu