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A Classification of Cyclic Nodes and Enumeration of Components of a Class of Discrete Graphs |
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PP: 103-112 |
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Author(s) |
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M. Khalid Mahmood,
Farooq Ahmad,
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Abstract |
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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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