I am not very familiar with graph theory, but I need some results for my work. Thus, the question is, whether the following has already been studied and where I can find it. Let $G=(V,E)$ be an graph and let $b$ be its first Betti number. We choose $b$ directed cycles in $G$, which form a homology basis. Further, we write $\tau_j\subseteq E$ for the set of edges in the $j$-th cycle. To every edge $e$ we associate an indeterminant $T_{e}$. Now consider the $b\times b$-matrix with entries $$a_{jk}=(-1)^{\epsilon(j,k)}\sum_{e\in\tau_j\cap\tau_k}T_e,$$ where $\epsilon(j,k)$ is $0$ if the $j$-th and the $k$-th cycle have the same direction on their intersection, and $1$ otherwise. I think, one can always choose the cycles such that $\epsilon(j,k)$ is always $0$.

Has a matrix like this been studied in graph theory, especially its determinant and its adjugate? This should be possible by elementary combinatoric, but it would be nice to have a reference and maybe there are more known interesting facts.