AI 中文总结
本文研究带不交环图的动力学，通过分裂变换得到标准形式，证明环长两两互素的3环流星图的四个等价条件成立，进而验证两个猜想对这类图成立。
AI 中文摘要
本文引入了标准形式的连通有限不交环图的概念，并证明任何此类图都可通过有限次内分裂和外分裂变换为标准形式图。据此，我们给出了长度为3的流星图强移位等价的数论判据，其中长度为3的流星图是由三个不交环构成的连通有限本质图，形成唯一的长度为3的环链。我们随后证明，环长两两互素的长度为3的流星图，其移位等价、强移位等价、对应的Leavitt路代数的分次Morita等价，以及分次K-理论$K^{gr}_0$作为$\boldsymbol{\text{Z}}[x, x^{-1}]$-模同构，这四个条件是等价的。因此，Williams猜想和Hazrat分次Morita等价猜想对于恰好包含三个环且环长两两互素的不交环图成立。
英文摘要
In this article, we introduce the notion of connected finite graphs with disjoint cycles in normal form and show that any such graph can be transformed into a normal form graph via a finite sequence of in-splittings and out-splittings. Consequently, we provide number-theoretic criteria for meteor graphs of length three to be strongly shift equivalent, where a meteor graph of length three is a connected finite essential graph consisting of three disjoint cycles which makes a unique chain of cycles of length three. We then prove that meteor graphs of length three whose cycle lengths are pairwise coprime are shift equivalent if and only if they are strongly shift equivalent, if and only if their corresponding Leavitt path algebras are graded Morita equivalent, if and only if their graded $K$-theories, $K^{gr}_0$, are order-preserving $\mathbb{Z}[x, x^{-1}]$-module isomorphic. As a consequence, Williams' Conjecture and Hazrat's Graded Morita Equivalence Conjecture hold for graphs with disjoint cycles that contain exactly three cycles whose lengths are pairwise coprime.
Commentsv2: Major revisions
