Cycle intersection form and oscillation of graph eigenfunctions

For a real symmetric matrix $H$ strictly supported on a finite simple graph, it is shown that the inertia of a weighted intersection form on the cycle space of the graph, with weights derived from a non-vanishing eigenvector of $H$, governs oscillation data on the graph.

Specifically, the null space of the form controls eigenvalue multiplicity, while its Morse index determines the number of sign changes across edges.

In the case of a simple eigenvalue, the Hessian at zero of the eigenvalue branch of the discrete magnetic Schrödinger operator is identified with the dual of the cycle intersection form.

Applications are given to stability analysis of coupled oscillator networks, to the local behavior of dispersion relations for (decorated) strained graphene, and to nodal-domain counts.