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Consider a graph G and its incident matrix $B\in R^{m\times n}$. We can compute the SVD of $B$ as $B=U^{\top}\Sigma V$. Note that the Laplacian matrix $L_G=B^{\top}B$, so the right singular vectors are the eigenvectors of $L_G$.

Now we are interested in the left singular vectors $(u_1,...,u_m)=U$. It seems that $(u_1,...,u_m)$ correspond to the set of edges in $G$, but I have no concrete idea on how they are related to each other.

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    $\begingroup$ What is an "interpretation" of a mathematical result for you? For instance, every locally-compact topological group admits and essentially unique Haar measure. How do you "interpret" this? $\endgroup$
    – Alex M.
    Commented Mar 17, 2021 at 21:48
  • $\begingroup$ Sorry, the question isn't phrased clearly. In fact, the question is asked because I have not known a rigorous result establishing the connection. Any reference or just intuition is welcome $\endgroup$
    – Bigtoe21
    Commented Mar 18, 2021 at 19:19
  • $\begingroup$ This matrix won't have a unique set of left singular vectors $U$ unless $m=n$ (and the eigenvalues of the Laplacian are distinct). This is pretty rare for graphs. The interesting singular vectors will just be $B$ applied to the eigenvectors of the Laplacian (up to scaling), and the remaining singular values are $0$. $\endgroup$
    – Will Sawin
    Commented Mar 21, 2021 at 15:27

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