AI 中文总结
研究是否存在非同构的度相似单圈图，通过符号叶剥离算法和彩色圈刚性引理证明不存在，与单圈图度相似的图必同构。
AI 中文摘要
如果两个图的邻接矩阵和度矩阵同时相似，则它们是度相似的。戈德西尔和孙询问是否存在非同构的度相似单圈图。我们证明不存在这样的图。每个与单圈图度相似的图都与其同构。证明使用了符号叶剥离算法来恢复附着在唯一圈上的有根树，并使用了彩色圈的刚性引理来识别圈的自同构。
英文摘要
Two graphs are degree-similar if their adjacency matrices and degree matrices are simultaneously similar. Godsil and Sun asked whether non-isomorphic degree-similar unicyclic graphs exist. We prove that they do not exist. Every graph degree-similar to a unicyclic graph is isomorphic to it. The proof uses a symbolic leaf-peeling algorithm to recover the rooted trees attached to the unique cycle, and a rigidity lemma for colored cycles to identify automorphisms of the cycle.
