Let $\mathcal{G}$ be an undirected graph and $M$ its corresponding oriented incidence matrix. Then, can the singular values of $M$ give an indication of how dense/sparse the graph? Particularly, the maximum and the minimum non-zero singular values of $M$?
2026-03-31 11:25:49.1774956349
Singular values of incidence matrix of a graph
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Yes. For a connected graph, the smallest non-zero singular value of $M$ is the square root of the graph's algebraic connectivity. The higher the value, the denser the graph.
I am not sure what can be said about the largest eigenvalue. That said, you might be able to find more using the fact that the squares of the singular values of the oriented incidence matrix are the square roots of the eigenvalues of the graph Laplacian.