Helmholzian spectra of graphs: basic properties

TL;DR

This paper explores the spectral properties of the Helmholtzian matrix of graphs, revealing its positive semi-definiteness, independence from orientation, and its relation to other graph spectra, with implications for various graph types.

Contribution

It introduces the Helmholtzian matrix as a new graph matrix, analyzes its spectral properties, and connects it to existing graph spectra, highlighting its role in bridging different graph classes.

Findings

Helmholtzian matrix is positive semi-definite.

Eigenvalues are independent of orientation.

Non-zero eigenvalues match those of the Laplacian.

Abstract

The Helmholtzian matrix of a graph $G=(V(G),E(G))$ is a graph-theoretic analogue of the vector Laplacian (or Helmholtz operator) [S. Li, L. Lu, J.F. Wang, A graph discretization of vector Laplacian, 379 (2026) 446--460]. Motivated by the applications of graph Helmholtzian in simplicial networks, we will investiagte its basic spectral properties. As the first graph matrix indexed by edge set, we find that Helmholtzian matrix is positive semi-definite and its non-negativity correlates with the odd cycles in $G$ and the orientation on $E(G)$, while its irreducibility relates to the signed graphs with loops. We show that the eigenvalues of Helmholtzian matrix are independent of the orientation and further investigate the eigenvalue interlacing under edge additions. One of striking findings is that the non-zero eigenvalues of the Laplacian matrix are those of Helmholtzian matrix of every graph. All these discoveries reveal that the Helmholtzian spectrum of $G$ balances and bridges the oriented graphs, weighted graphs and signed graphs as well as their adjacency or Laplacian spectra.