The Helmholtzian matrix on graphs admits a classification of graphs with two eigenvalues, a formula for its nullity, and a combinatorial interpretation of its polynomial coefficients.
Huang, Induced subgraphs of hypercubes and a proof of the Sensitivity Conjecture, Ann
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Helmholzian Spectra of Graphs: Novel Properties
The Helmholtzian matrix on graphs admits a classification of graphs with two eigenvalues, a formula for its nullity, and a combinatorial interpretation of its polynomial coefficients.