Extend the oscillation formula to eigenvectors with zeros and to multiple eigenvalues

Extend the nodal oscillation formula for real symmetric matrices strictly supported on a finite connected graph G—namely, that for a simple eigenvalue λk with eigenvector ψ having no zero entries the nodal count (# of directed edges (r→s) with ψr Hrs ψs > 0) equals k − 1 plus the Morse index of the weighted cycle intersection form [Φ−1]Z, where Φ is the diagonal operator on directed edges with entries −ψr Hrs ψs and Z is the cycle subspace—to the cases where the eigenvector ψ has zero entries and/or the eigenvalue λk has multiplicity greater than one.

Background

The main theorem proves an exact oscillation identity for the nodal count of an eigenvector ψ corresponding to a simple eigenvalue and with no zero entries, linking excess sign changes to the Morse index of a weighted cycle intersection form. This yields a precise formula expressing the nodal count as k−1 plus the number of negative eigenvalues of the compressed inverse Φ−1 on the cycle space.

However, several works on graph nodal structures also consider eigenfunctions with zeros and non-simple (degenerate) eigenvalues. The present result assumes simplicity and non-vanishing entries; extending the formula to include zero entries of ψ and eigenvalue multiplicity is identified as an unresolved problem.