I believe you are looking for a sparse Cholesky factorization (performed after column-row pivoting). In this sense, the view of the graph begins to intertwine with a sparse-matrix interpretation. The graph cycle appears as a "feedback loop", in the linear operator interpretation of Laplacian matrix, and is eliminated by Gaussian elimination.

It is worth noting that for a tree, the Cholesky factorization coincides with the oriented incidence matrix of the corresponding graph, assuming that factorization is performed in the correct order, from the root to the leaves. For graphs containing many cycles, e.g. the grid graph, the factorizations are fully dense, despite originating from a sparse matrix. For everything in-between, the density of the factorization is a good gauge to how interconnected the graph is.

I'm happy to provide refs / more info / detail if you can elaborate on exactly what you're looking for.

I believe you are looking for a sparse Cholesky factorization (performed after column-row pivoting). In this sense, the view of the graph begins to intertwine with a sparse-matrix interpretation. The graph cycle appears as a "feedback loop", in the linear operator interpretation of Laplacian matrix, and is eliminated by Gaussian elimination.

It is worth noting that for a tree, the Cholesky factorization coincides with the oriented incidence matrix of the corresponding graph, assuming that factorization is performed in the correct order, from the leaves to the root (EDIT: corrected!). For graphs containing many cycles, e.g. the grid graph, the factorizations are fully dense, despite originating from a sparse matrix. For everything in-between, the density of the factorization is a good gauge to how interconnected the graph is.

I'm happy to provide refs / more info / detail if you can elaborate on exactly what you're looking for.