%0 Journal Article
%T Loops in Digraphs of Lambert  Mapping Modulo Prime Powers:  Enumerations and Applications
%A M. Khalid Mahmood
%A Lubna Anwar
%J Advances in Pure Mathematics
%P 564-570
%@ 2160-0384
%D 2016
%I Scientific Research Publishing
%R 10.4236/apm.2016.68045
%X For an odd prime number p, and positive integers k and <img src=\"Edit_4f0a5616-a1b0-43ff-955c-7901d28b836a.bmp\" alt=\"\" />, we denote <img src=\"Edit_2b486784-22dc-4d21-8289-af0969e92aa7.bmp\" alt=\"\" />, a digraph for which&nbsp;<img src=\"Edit_a196c1a3-6439-4c6d-b661-b0a90b5f6c54.bmp\" alt=\"\" /> is the set of vertices and there is a directed edge from <em>u</em> to <em>v</em> if <img src=\"Edit_41881c8d-42dc-473a-beba-15ad45599a53.bmp\" alt=\"\" />, where &nbsp;<img src=\"Edit_7f2413fd-bcf6-47a3-959b-2c7f0c7db93a.bmp\" alt=\"\" />. In this work, we study isolated and non-isolated fixed points (or loops) in digraphs arising from Discrete Lambert Mapping. It is shown that if <img src=\"Edit_fdd83257-d177-45b3-9fb7-657c266ea5f9.bmp\" alt=\"\" />, then all fixed points in <img src=\"Edit_3f3ea701-cea1-46ff-b22c-a9b7803a0427.bmp\" alt=\"\" />&nbsp;are isolated. It is proved that the digraph <img src=\"Edit_8ea79992-53ca-46df-adf7-3cf1a6e9076b.bmp\" alt=\"\" />&nbsp;has <img src=\"Edit_1c6ae783-036f-4d62-a3de-0184c7acb8f9.bmp\" alt=\"\" />isolated fixed points only if <img src=\"Edit_78cec9a2-2a12-4082-a840-f292737c6e0f.bmp\" alt=\"\" />. It has been characterized that <img src=\"Edit_dbbc1742-deb8-450a-baa2-115029a7f94b.bmp\" alt=\"\" />&nbsp;has no cycles except fixed points if and only if either <em>g</em> is of order 2 or <em>g</em> is divisible by <em>p</em>. As an application of these loops, the solvability of the exponential congruence <img src=\"Edit_dd78132a-3fc7-445a-921d-06bd448164ff.bmp\" alt=\"\" />&nbsp;has been discussed.
%K Fixed Points
%K Lambert Map
%K Multiplicative Order
%U http://www.scirp.org/journal/PaperInformation.aspx?PaperID=69215