Orderly Algorithm to enumerate central groupoids and their graphs
Tim Boykett
Abstract
A graph has the unique path property UPPn if there is a unique path of length n between any ordered pair of nodes. This paper reiterates Royle and MacKay's technique for constructing orderly algorithms. We wish to use this technique to enumerate all UPP2 graphs of small orders 9 and 16. We attempt to use the direct graph formalism and find that the algorithm is inefficient. We introduce a generalised problem and derive algebraic and combinatoric structures with appropriate structure. We are able to then design an orderly algorithm to determine all UPP2 graphs of order 9, which runs fast enough. We hope to be able to determine the UPP2 graphs of order 16 in the near future.
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