Cycle space

Cycle space — пространство циклов.

Given a graph $\,G$ , let $\,e_{1}, e_{2}, \ldots, e_{|E(G)|}$ be an ordering of its edges. Then a subset $\,S$ of $\,E(G)$ corresponds to a $\,(0,1)$ -vector $\,(b_{1}, \ldots, b_{|E(G)|})$ in the usual way with $\,b_{i}= 1$ if $\,e_{i} \in S$ , and $\,b_{i} = 0$ if $\,e_{i} \not \in S$ . These vectors form an $|E(G)|$ -dimensional vector space, denoted by $(Z_{2})^{|E(G)|}$ , over the field of integers modulo $\,2$ . The vectors in $\,(Z_{2})^{|E(G)|}$ which correspond to the cycles in $\,G$ generate a subspace called the cycle space of $\,G$ denoted by $\,{\mathcal C}(G)$ . It is known that