Gluing Formula for the Pseudo-Determinant of Graph Laplacian and Applications to Counting of Spanning Trees

What the math gives to ML

The paper provides exact determinant and spanning-tree identities for gluing graph pieces along a shared boundary, expressed through the boundary Dirichlet-to-Neumann operator and the pseudo-determinant of graph Laplacians. The transferable asset is compositional spectral elimination: an internal subgraph can be replaced by a small boundary response matrix while preserving its effect on signals entering through the boundary. This suggests hierarchical graph neural networks whose pooling and message passing operate on learned boundary interfaces rather than all vertices, together with connectivity regularizers based on differentiable Laplacian pseudo-determinants. The strongest closed forms target unweighted simple graphs, so weighted neural implementations should validate the extension empirically.