图的拉普拉斯矩阵特征向量的节点边数，是指该特征向量在其上发生变号的边的数量。这一量可推广到任何支撑在具有 n 个顶点的图 G 上的实对称 n×n 矩阵。已知，对给定矩阵的所有特征向量取平均得到的平均节点数介于 (n−1)/2 与 (n−1)/2+β(G) 之间，其中 β(G) 是 G 的第一贝蒂数（一个拓扑量）；此前人们认为，在一般情形下，该平均值应在 (n−1)/2+β(G)/2 附近。我们证明事实并非如此：该平均值介于 (n−1)/2+β(G)/n 与 (n−1)/2+β(G)−β(G)/n 之间，并且对于 n 和 β 的任何可能选择，我们都给出了达到上界和下界的图与矩阵。定义节点数时，对矩阵的一个自然条件是其具有单重特征值且特征向量处处非零。对于任意连通图 G，支撑在 G 上的一般实对称矩阵都满足这一节点数条件。然而，常数对角矩阵的情形要微妙得多。我们完整刻画了使该条件在一般意义下成立的图 G，并表明若不成立，则任何支撑在 G 上、具有常数对角元的实对称矩阵都有一个多重特征值，或有一个在某处消失的特征向量。最后，我们讨论了当这一节点数条件失效时可以得出的结论，并给出了示例。
英文摘要：
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 $\frac{n-1}{2}+β(G)$, where $β(G)$ is the first Betti number of $G$ (a topological quantity), and it was believed that generically the average should be around $\frac{n-1}{2}+β(G)/2$. We prove that this is not the case: the average is bounded between $\frac{n-1}{2}+β(G)/n$ and $\frac{n-1}{2}+β(G)-β(G)/n$, 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.
