Average Nodal Count and the Nodal Count Condition for Graphs

The nodal edge count of an eigenvector of the Laplacian of a graph is the number of edges on which it changes sign. This quantity extends to any real symmetric $n\times n$ matrix supported on a graph $G$ with $n$ vertices. The average nodal count, averaged over all eigenvectors of a given matrix, is known to be bounded between $\frac{n-1}{2}$ and $\frac{n-1}{2}+\beta(G)$, where $\beta(G)$ is the first Betti number of $G$ (a topological quantity), and it was believed that generically the average should be around $\frac{n-1}{2}+\beta(G)/2$. We prove that this is not the case: the average is bounded between $\frac{n-1}{2}+\beta(G)/n$ and $\frac{n-1}{2}+\beta(G)-\beta(G)/n$, and we provide graphs and matrices that attain the upper and lower bounds for any possible choice of $n$ and $\beta$. A natural condition on a matrix for defining the nodal count is that it has simple eigenvalues and non-vanishing eigenvectors. For any connected graph $G$, a generic real symmetric matrix supported on $G$ satisfies this nodal count condition. However, the situation for constant diagonal matrices is far more subtle. We completely characterize the graphs $G$ for which this condition is generically true, and show that if this is not the case, then any real symmetric matrix supported on $G$ with constant diagonal has a multiple eigenvalue or an eigenvector that vanishes somewhere. Finally, we discuss what can be said when this nodal count condition fails, and provide examples.