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× 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 n-1/2 and n-1/2+β(G), where β(G) is the first Betti number of G (a topological quantity), and it was believed that generically the average should be around n-1/2+β(G)/2. We prove that this is not the case: the average is bounded between n-1/2+β(G)/n and n-1/2+β(G)-β(G)/n, and we provide graphs and matrices that attain the upper and lower bounds for any possible choice of n and β. 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.