Regularity of Congruential Graphs

The aim of this article is to make a link between the congruential systems investigated by Conway and the infinite graphs theory. We compare the graphs of congruential systems with a well known family of infinite graphs: the regular graphs of finite degree considered by Muller and Shupp, and by Courcelle. We first consider congruential systems as word rewriting systems to extract some subfamilies of congruential systems, the q-p-congruential systems, representing the regular graphs of finite degree. We then prove the non-regularity of the Collatz's graph.

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