基于等谱约简的共谱顶点特征化
Characterizing cospectral vertices via isospectral reduction
摘要：
在图论中，两个新兴的主题是研究图的共谱顶点和研究图的等谱约简。在本文中，我们证明了这两个领域之间的一个基本关系，这是一个图的两个顶点是共谱的当且仅当这些顶点上的等谱约简有一个非平凡的自同构。众所周知，如果图的两个顶点是对称的，即如果存在置换这两个顶点的图自同构，则它们是共谱的。本文推广了这一结果，证明了任意两个共谱顶点在图的某些简化版本中是对称的。我们还证明了两个顶点是强共谱的当且仅当它们是共谱的，并且这两个顶点上的等谱约化具有简单的特征值。我们进一步描述了如何使用这些结果来构建新的家庭的图的共谱顶点。
英文摘要：
Two emerging topics in graph theory are the study of cospectral vertices of a graph, and the study of isospectral reductions of graphs. In this paper, we prove a fundamental relationship between these two areas, which is that two vertices of a graph are cospectral if and only if the isospectral reduction over these vertices has a nontrivial automorphism. It is well known that if two vertices of a graph are symmetric, i.e. if there exists a graph automorphism permuting these two vertices, then they are cospectral. This paper extends this result showing that any two cospectral vertices are symmetric in some reduced version of the graph. We also prove that two vertices are strongly cospectral if and only if they are cospectral and the isospectral reduction over these two vertices has simple eigenvalues. We further describe how these results can be used to construct new families of graphs with cospectral vertices.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
