The Helmholtzian matrix is positive semi-definite, its eigenvalues are independent of edge orientation, and its non-zero eigenvalues coincide exactly with those of the ordinary graph Laplacian for every graph.
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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.
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Helmholzian spectra of graphs: basic properties
The Helmholtzian matrix is positive semi-definite, its eigenvalues are independent of edge orientation, and its non-zero eigenvalues coincide exactly with those of the ordinary graph Laplacian for every graph.
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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.