Applied Mathematics & Information Sciences

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Let Zn be the ring of residue classes modulo n. Define f : Zn 7→ Zn by f (x) = x4. Action of this map is studied by means of digraphs which produce an edge from the residue classes a to b if f (a) ≡ b. For every integer n, an explicit formula is given for the number of fixed points of f . It is shown that the graph G(pk), k ≥ 1 has four fixed points if and only if 3 | p−1 and has two fixed points if and only if 3 ∤ p−1. A classification of cyclic vertices of the graph G(pk) has been determined. A complete enumeration of non-isomorphic cycles and components of G(pk) has been explored.

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