Helmholtzian spectra of graphs: Basic properties

The Helmholtzian matrix of a graph G = ( V ( G ) , E ( G ) ) is a graph-theoretic analogue of the vector Laplacian (or Helmholtz operator) introduced in S. Li, L. Lu and J. F. Wang, A graph discretization of vector Laplacian , Discrete Appl. Math. 379 (2026), 446–460. Motivated by applications of the graph Helmholtzian to simplicial networks, we investigate its basic spectral properties. The matrix is naturally indexed by the edges of G and is positive semidefinite for every choice of orientations. We characterize when an edge orientation makes it entrywise nonnegative and relate its irreducibility to an associated signed graph with loops. We prove that the Helmholtzian spectrum is independent of the edge and triangle orientations and establish eigenvalue interlacing inequalities for induced subgraphs. We also show that the nonzero Helmholtzian spectrum, with multiplicities, is the multiset union of the nonzero Laplacian spectrum of G and the nonzero spectrum of a shifted adjacency matrix of the signed graph on the triangles of G . These results connect the Helmholtzian spectrum with ordinary Laplacian spectra, signed graphs, and combinatorial Hodge theory.