The Iteration Digraphs of Lambert Map Over the Local Ring Z/pkZ: Structures and Enumerations

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Khalid Mahmood M.; Anwar Lubna

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Journal Paper

Paper Information

Journal: Iranian Journal of Mathematical Sciences and Informatics Year:2022 | Volume:17 | Issue:2 Page(s): 307-314

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Title

The Iteration Digraphs of Lambert Map Over the Local Ring Z/pkZ: Structures and Enumerations

Author(s)

Khalid Mahmood M. | Anwar Lubna | Issue Writer Certificate

Keywords

Fixed points

Lambert map

Multiplicative order

Abstract

Let p be prime and , : x 7! xgx, the Discrete Lambert map. For k ,1, let V = {0, 1, 2, . . ., pk −,1}. The iteration digraph is a directed graph with V as the vertex set and there is a unique directed edge from u to , (u) for each u 2 V. We denote this digraph by G(g, pk), where g 2 (Z/pkZ), . In this piece of work, we investigate the structural properties and find new results modulo higher powers of primes. We show that if g is of order pd, 1 ,d ,k −,1 then G(g, pk) has pk−, d d 2 e loops. If g = tp + 1, 1 ,t ,pk−, 1 −,1 then the digraph contains p k +1 2 cycles. Further, if g has order pk−, 1 then G(g, pk) has p −,1 cycles of length pk−, 1 and the digraph is cyclic. We also propose explicit formulas for the enumeration of components.

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APA: Copy

Khalid Mahmood, M., & Anwar, Lubna. (2022). The Iteration Digraphs of Lambert Map Over the Local Ring Z/pkZ: Structures and Enumerations. IRANIAN JOURNAL OF MATHEMATICAL SCIENCES AND INFORMATICS (IJMSI), 17(2), 307-314. SID. https://sid.ir/paper/1004754/en

Vancouver: Copy

Khalid Mahmood M., Anwar Lubna. The Iteration Digraphs of Lambert Map Over the Local Ring Z/pkZ: Structures and Enumerations. IRANIAN JOURNAL OF MATHEMATICAL SCIENCES AND INFORMATICS (IJMSI)[Internet]. 2022;17(2):307-314. Available from: https://sid.ir/paper/1004754/en

IEEE: Copy

M. Khalid Mahmood, and Lubna Anwar, “The Iteration Digraphs of Lambert Map Over the Local Ring Z/pkZ: Structures and Enumerations,” IRANIAN JOURNAL OF MATHEMATICAL SCIENCES AND INFORMATICS (IJMSI), vol. 17, no. 2, pp. 307–314, 2022, [Online]. Available: https://sid.ir/paper/1004754/en

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