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.



A certain matrix associated to graphs
For a connected $G$, in the case of the cycles being chosen in the usual way --- one for each edge of $G$ outside a spanning tree $\Theta$ of $G$, this matrix can be identified with a submatrix of $A^\top A$, for $A$ the oriented [incidence matrix](http://en.wikipedia.org/wiki/Incidence_matrix) of $G$. That is, $A$ is an $|V|\times |E|$ matrix of the map $\mathbb{R}E\to \mathbb{R}V$ given by
$$
A\_{ve}=\begin{cases} 0 & v\not\in e\\ 1 &e=(vv')\\-1 &e=(v'v)\end{cases}.
$$
Choosing the ordering of $E$ so that the edges outside $\Theta$ come first, the matrix in question is the $b\times b$ submatrix of $A^\top A$ formed by its 1st $b$ rows and columns.


Note that $A^\top A-2I$ is some kind of $1,-1$-adjacency matrix of the directed line graph of $G$.


