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Let $G$ be a connected unweighted undirected graph. In addition, let $\lambda_2(L(G))$ be the second smallest eigenvalue of the Laplacian matrix of graph $G$.

Is $\lambda_2(L(G))$ a submodular set function over the edges (for a fixed number of nodes)?

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    $\begingroup$ Have you checked this numerically for small cases (all graphs with $v\leq 5$, graphs near Petersen and other standard examples, etc.)? $\endgroup$
    – Noam D. Elkies
    Commented Jan 16, 2017 at 20:53

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