Enumerative combinatorics
Nowhere-zero integral chains and flows in bidirected graphs
Journal of Combinatorial Theory Series B
European Journal of Combinatorics
Handbook of combinatorics (vol. 1)
Polynomials associated with nowhere-zero flows
Journal of Combinatorial Theory Series B
Biased graphs IV: geometrical realizations
Journal of Combinatorial Theory Series B
Counting Integer Flows in Networks
Foundations of Computational Mathematics
The flow and tension spaces and lattices of signed graphs
European Journal of Combinatorics
Bounds on the coefficients of tension and flow polynomials
Journal of Algebraic Combinatorics: An International Journal
Enumerating colorings, tensions and flows in cell complexes
Journal of Combinatorial Theory Series A
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A nowhere-zero k-flow on a graph @C is a mapping from the edges of @C to the set {+/-1,+/-2,...,+/-(k-1)}@?Z such that, in any fixed orientation of @C, at each node the sum of the labels over the edges pointing towards the node equals the sum over the edges pointing away from the node. We show that the existence of an integral flow polynomial that counts nowhere-zero k-flows on a graph, due to Kochol, is a consequence of a general theory of inside-out polytopes. The same holds for flows on signed graphs. We develop these theories, as well as the related counting theory of nowhere-zero flows on a signed graph with values in an abelian group of odd order. Our results are of two kinds: polynomiality or quasipolynomiality of the flow counting functions, and reciprocity laws that interpret the evaluations of the flow polynomials at negative integers in terms of the combinatorics of the graph.