Helmholzian Spectra of Graphs: Novel Properties

TL;DR

This paper explores the spectral properties of the graph Helmholtzian matrix, providing new theoretical insights, classifications, and bounds related to its eigenvalues and graph structures.

Contribution

It offers a new proof that the Helmholtzian matrix represents the graph Helmholtzian and investigates its spectral characteristics and classifications.

Findings

Classified graphs with exactly two Helmholtzian eigenvalues

Determined the nullity of the Helmholtzian matrix

Provided bounds for the smallest Helmholtzian eigenvalue

Abstract

Let $\grad$, $\curl$, and $\dv$ be the graph-theoretic analogues of the gradient, curl, and divergence operators from multivariate calculus. The graph Laplacian $-\dv \grad$ gives rise to the celebrated Laplacian matrix, while the matrix representation of the graph Helmholtzian $\grad \grad^* + \curl^* \curl$ is called the Helmholtzian matrix. In this paper, we present a new graph-theoretic proof that the Helmholtzian matrix indeed represents the graph Helmholtzian. We then investigate the spectral properties of this matrix. Our main results are as follows: (i) a classification of graphs having exactly two distinct Helmholtzian eigenvalues; (ii) the nullity of the Helmholtzian matrix; and (iii) a combinatorial interpretation of the coefficients of the Helmholtzian polynomial. Furthermore, we determine the Helmholtzian spectrum for certain graph products and characterize Helmholtzian integral graphs, as well as derive bounds for the smallest Helmholtzian eigenvalue. Meanwhile, we pose some open problems for future research.