Williams' and Graded Equivalence Conjectures for small graphs

TL;DR

This paper proves that for small graphs with three vertices, the Williams and Graded Equivalence Conjectures hold, establishing equivalences between shift equivalence, strong shift equivalence, and Morita equivalence of associated algebras.

Contribution

It confirms the Williams and Graded Equivalence Conjectures for a specific class of small graphs, linking graph dynamics and algebraic equivalences.

Findings

Williams Conjecture holds for small graphs.

Graded Morita Equivalence Conjecture holds for small graphs.

Equivalence of shift and Morita equivalences in this class.

Abstract

We prove what might have been expected: The Williams Conjecture in symbolic dynamics and Graded Morita Equivalence Conjecture for Leavitt/$C^*$-graph algebras hold for ``small graphs'', i.e., connected graphs with three vertices, no parallel edges, no sinks with no trivial hereditary and saturated subsets. Namely, two small graphs are shift equivalent if and only if they are strong shift equivalent if and only if their Leavitt/$C^*$-graph algebras are graded/equivariant Morita equivalent.