Williams' and Graded Equivalence Conjectures for small graphs

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.