Nowhere-zero Flow
In graph theory, a nowhere-zero flow or NZ flow is a network flow that is nowhere zero. It is intimately connected (by duality) to coloring planar graphs. Definitions Let ''G'' = (''V'',''E'') be a digraph and let ''M'' be an abelian group. A map ''φ'': ''E'' → ''M'' is an ''M''-circulation if for every vertex ''v'' ∈ ''V'' :\sum_ \phi(e) = \sum_ \phi(e), where ''δ''+(''v'') denotes the set of edges out of ''v'' and ''δ''−(''v'') denotes the set of edges into ''v''. Sometimes, this condition is referred to as Kirchhoff's law. If ''φ''(''e'') ≠ 0 for every ''e'' ∈ ''E'', we call ''φ'' a nowhere-zero flow, an ''M''-flow, or an NZ-flow. If ''k'' is an integer and 0 < , ''φ''(''e''), < ''k'' then ''φ'' is a ''k''-flow.

Other notions

Let ''G'' = (''V'',''E'') be an . An ori ...
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Graph Theory
In mathematics, graph theory is the study of ''graphs'', which are mathematical structures used to model pairwise relations between objects. A graph in this context is made up of '' vertices'' (also called ''nodes'' or ''points'') which are connected by '' edges'' (also called ''links'' or ''lines''). A distinction is made between undirected graphs, where edges link two vertices symmetrically, and directed graphs, where edges link two vertices asymmetrically. Graphs are one of the principal objects of study in discrete mathematics. Definitions Definitions in graph theory vary. The following are some of the more basic ways of defining graphs and related mathematical structures. Graph In one restricted but very common sense of the term, a graph is an ordered pair G=(V,E) comprising: * V, a set of vertices (also called nodes or points); * E \subseteq \, a set of edges (also called links or lines), which are unordered pairs of vertices (that is, an edge is associated with t ...
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