Cycle space

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In graph theory, a branch of mathematics, the (binary) cycle space of an undirected graph is the set of its even-degree spanning subgraphs, or the set of their edge sets. This set of subgraphs can be described algebraically as a vector space over Z 2 {\displaystyle \mathbb {Z} _{2}} (the field with two elements). The dimension of this space is the circuit rank, or cyclomatic number, of the graph. The same space can also be described in terms from algebraic topology as the first homology group of the graph. Using homology theory, the binary cycle space may be generalized to cycle spaces over arbitrary rings.