Dynamics on graphs with disjoint cycles and applications

Abstract

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.