Edge space

In the mathematical discipline of graph theory, the edge space and vertex space of an undirected graph are vector spaces defined in terms of the edge and vertex sets, respectively. These vector spaces make it possible to use techniques of linear algebra in studying the graph. In the mathematical discipline of graph theory, the edge space and vertex space of an undirected graph are vector spaces defined in terms of the edge and vertex sets, respectively. These vector spaces make it possible to use techniques of linear algebra in studying the graph. Let G := ( V , E ) {displaystyle G:=(V,E)} be a finite undirected graph. The vertex space V ( G ) {displaystyle {mathcal {V}}(G)} of G is the vector space over the finite field of two elements Z / 2 Z := { 0 , 1 } {displaystyle mathbb {Z} /2mathbb {Z} :=lbrace 0,1 brace } of all functions V → Z / 2 Z {displaystyle V ightarrow mathbb {Z} /2mathbb {Z} } . Every element of V ( G ) {displaystyle {mathcal {V}}(G)} naturally corresponds the subset of V which assigns a 1 to its vertices. Also every subset of V is uniquely represented in V ( G ) {displaystyle {mathcal {V}}(G)} by its characteristic function. The edge space E ( G ) {displaystyle {mathcal {E}}(G)} is the Z / 2 Z {displaystyle mathbb {Z} /2mathbb {Z} } -vector space freely generated by the edge set E. The dimension of the vertex space is thus the number of vertices of the graph, while the dimension of the edge space is the number of edges.