{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,3,13]],"date-time":"2026-03-13T13:07:25Z","timestamp":1773407245482,"version":"3.50.1"},"reference-count":9,"publisher":"Wiley","issue":"1","license":[{"start":{"date-parts":[[2012,10,3]],"date-time":"2012-10-03T00:00:00Z","timestamp":1349222400000},"content-version":"vor","delay-in-days":276,"URL":"http:\/\/creativecommons.org\/licenses\/by\/3.0\/"}],"content-domain":{"domain":["onlinelibrary.wiley.com"],"crossmark-restriction":true},"short-container-title":["Journal of Applied Mathematics"],"published-print":{"date-parts":[[2012,1]]},"abstract":"<jats:p>We introduce and investigate a new class of graphs arrived from exponential congruences. For each pair of positive integers<jats:italic>a<\/jats:italic>and<jats:italic>b<\/jats:italic>, let<jats:italic>G<\/jats:italic>(<jats:italic>n<\/jats:italic>) denote the graph for which<jats:italic>V<\/jats:italic>= {0, 1, \u2026,<jats:italic>n<\/jats:italic>\u2212 1} is the set of vertices and there is an edge between<jats:italic>a<\/jats:italic>and<jats:italic>b<\/jats:italic>if the congruence<jats:italic>a<\/jats:italic><jats:sup><jats:italic>x<\/jats:italic><\/jats:sup>\u2261<jats:italic>b<\/jats:italic>\u2009(mod\u2009<jats:italic>n<\/jats:italic>) is solvable. Let be the prime power factorization of an integer<jats:italic>n<\/jats:italic>, where<jats:italic>p<\/jats:italic><jats:sub>1<\/jats:sub>&lt;<jats:italic>p<\/jats:italic><jats:sub>2<\/jats:sub>&lt; \u22ef&lt;<jats:italic>p<\/jats:italic><jats:sub><jats:italic>r<\/jats:italic><\/jats:sub>are distinct primes. The number of nontrivial self\u2010loops of the graph<jats:italic>G<\/jats:italic>(<jats:italic>n<\/jats:italic>) has been determined and shown to be equal to . It is shown that the graph<jats:italic>G<\/jats:italic>(<jats:italic>n<\/jats:italic>) has 2<jats:sup><jats:italic>r<\/jats:italic><\/jats:sup>components. Further, it is proved that the component \u0393<jats:sub><jats:italic>p<\/jats:italic><\/jats:sub>of the simple graph<jats:italic>G<\/jats:italic>(<jats:italic>p<\/jats:italic><jats:sup>2<\/jats:sup>) is a tree with root at zero, and if<jats:italic>n<\/jats:italic>is a Fermat\u2032s prime, then the component \u0393<jats:sub><jats:italic>\u03d5<\/jats:italic>(<jats:italic>n<\/jats:italic>)<\/jats:sub>of the simple graph<jats:italic>G<\/jats:italic>(<jats:italic>n<\/jats:italic>) is complete.<\/jats:p>","DOI":"10.1155\/2012\/292895","type":"journal-article","created":{"date-parts":[[2012,10,3]],"date-time":"2012-10-03T21:03:59Z","timestamp":1349298239000},"update-policy":"https:\/\/doi.org\/10.1002\/crossmark_policy","source":"Crossref","is-referenced-by-count":6,"title":["On Simple Graphs Arising from Exponential Congruences"],"prefix":"10.1155","volume":"2012","author":[{"given":"M. Aslam","family":"Malik","sequence":"first","affiliation":[]},{"given":"M. Khalid","family":"Mahmood","sequence":"additional","affiliation":[]}],"member":"311","published-online":{"date-parts":[[2012,10,3]]},"reference":[{"key":"e_1_2_7_1_2","doi-asserted-by":"publisher","DOI":"10.1016\/j.ipl.2008.05.002"},{"key":"e_1_2_7_2_2","doi-asserted-by":"crossref","first-page":"229","DOI":"10.1080\/00150517.1998.12428933","article-title":"Power digraphs modulo n","volume":"36","author":"Wilson B.","year":"1998","journal-title":"The Fibonacci Quarterly"},{"key":"e_1_2_7_3_2","doi-asserted-by":"publisher","DOI":"10.1023\/B:CMAJ.0000042385.93571.58"},{"key":"e_1_2_7_4_2","doi-asserted-by":"publisher","DOI":"10.4236\/ojdm.2011.13013"},{"key":"e_1_2_7_5_2","doi-asserted-by":"publisher","DOI":"10.1016\/j.disc.2011.11.007"},{"key":"e_1_2_7_6_2","volume-title":"Graphs and Digraphs,","author":"Chartrand G.","year":"1996"},{"key":"e_1_2_7_7_2","volume-title":"The Theory of Numbers","author":"Adler A.","year":"1995"},{"key":"e_1_2_7_8_2","volume-title":"Discrete Mathematics and its Application","author":"Rosen K. H.","year":"1999"},{"key":"e_1_2_7_9_2","volume-title":"Elementry Numbers Theory with Applications","author":"Koshy T.","year":"2007"}],"container-title":["Journal of Applied Mathematics"],"original-title":[],"language":"en","link":[{"URL":"http:\/\/downloads.hindawi.com\/journals\/jam\/2012\/292895.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"http:\/\/downloads.hindawi.com\/journals\/jam\/2012\/292895.xml","content-type":"application\/xml","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/onlinelibrary.wiley.com\/doi\/pdf\/10.1155\/2012\/292895","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,4,9]],"date-time":"2025-04-09T22:46:41Z","timestamp":1744238801000},"score":1,"resource":{"primary":{"URL":"https:\/\/onlinelibrary.wiley.com\/doi\/10.1155\/2012\/292895"}},"subtitle":[],"editor":[{"given":"Maurizio","family":"Porfiri","sequence":"additional","affiliation":[]}],"short-title":[],"issued":{"date-parts":[[2012,1]]},"references-count":9,"journal-issue":{"issue":"1","published-print":{"date-parts":[[2012,1]]}},"alternative-id":["10.1155\/2012\/292895"],"URL":"https:\/\/doi.org\/10.1155\/2012\/292895","archive":["Portico"],"relation":{},"ISSN":["1110-757X","1687-0042"],"issn-type":[{"value":"1110-757X","type":"print"},{"value":"1687-0042","type":"electronic"}],"subject":[],"published":{"date-parts":[[2012,1]]},"assertion":[{"value":"2012-03-19","order":0,"name":"received","label":"Received","group":{"name":"publication_history","label":"Publication History"}},{"value":"2012-09-03","order":1,"name":"accepted","label":"Accepted","group":{"name":"publication_history","label":"Publication History"}},{"value":"2012-10-03","order":2,"name":"published","label":"Published","group":{"name":"publication_history","label":"Publication History"}}],"article-number":"292895"}}