Oscillation of graph eigenfunctions

An oscillation formula is established for the $k$-th eigenvector (assumed to be simple and with non-zero entries) of a weighted graph operator. The formula directly attributes the number of sign changes exceeding $k-1$ to the cycles in the graph, by identifying it as the Morse index of a weighted cycle intersection form introduced in the text. Two proofs are provided for the main result. Additionally, it is related to the nodal–magnetic theorem of Berkolaiko and Colin de Verdière and to a similar identity of Bronski, DeVille and Ferguson obtained for the linearization of coupled oscillator network equations around a known solution.