From Wikipedia:
The cylce space of an undirected graph is the set of its Eulerian subgraphs. This set of subgraphs can be described algebraically as a vector space over the two-element finite field.
The vector addition operation is the symmetric difference.
My main question is if the number of independent even cycles in a graph is well-defined or if it is dependent on the choice of cycles.
It occurred to me that if the subset of even cycles is closed under vector addition, then it would form a subspace of the cycle space and the number of independent even cycles is the unique value that is the dimension of that subspace.
I've found a reference (Theorem 2.2.9) that states
The multiplicity of the eigenvalue ${}-2$ in the line graph $L(H)$ is equal to the maximal number of independent even cycles in $H$.
Is this maximal number just the dimension of the basis or is such a basis not even well-defined.
Let $\mathscr{C}_0(G)=\{E\in\mathscr{C}(G):|E|\text{ is even}\}$. Let $E_0,E_1\in\mathscr{C}_0(G)$, with $|E_0|=2m$ and $|E_1|=2n$. If $k=|E_0\cap E_1|$, then
$$|E_0\mathbin{\triangle}E_1|=|E_0\setminus E_1|+|E_1\setminus E_0|=(2m-k)+(2n-k)=2(m+n-k)\;,$$
so $E_0\mathbin{\triangle}E_1\in\mathscr{C}_0(G)$, and $\mathscr{C}_0(G)$ is indeed a subspace of $\mathscr{C}(G)$.