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arXiv · 2607.16737

Degree-similar Unicyclic Graphs are Isomorphic

Yi-Liu Zhang·Yi-Zheng Fan
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Abstract

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.

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Yi-Liu Zhang, Yi-Zheng Fan. 2026-07-18. Degree-similar Unicyclic Graphs are Isomorphic. https://arxiv.org/abs/2607.16737

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