Analyzing nodal domains is a way to discern the structure of eigenvectors of operators on a graph. We give a new definition extending the concept of nodal domains to arbitrary signed graphs, and therefore to arbitrary symmetric matrices. We show that for an arbitrary symmetric matrix, a positive fraction of eigenbases satisfy a generalized version of known nodal bounds for un-signed (that is classical) graphs. We do this through an explicit decomposition. Moreover, we show that with high probability, the number of nodal domains of a bulk eigenvector of the adjacency matrix of a signed Erdős-Rényi graph is $\Omega (n/\log n)$ and $o(n)$.

中文翻译：

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对称矩阵的节点分解

分析节点域是识别图上算子特征向量结构的一种方法。我们给出了一个新的定义，将节点域的概念扩展到任意符号图，从而扩展到任意对称矩阵。我们证明，对于任意对称矩阵，特征基的正分数满足无符号（即经典）图的已知节点边界的广义版本。我们通过显式分解来做到这一点。此外，我们表明，有符号的 Erdős-Rényi 图的邻接矩阵的体特征向量的节点域数量很可能为 $\Omega (n/\log n)$ 和 $o(n)$。