{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2023,6,23]],"date-time":"2023-06-23T04:35:51Z","timestamp":1687494951430},"reference-count":9,"publisher":"World Scientific Pub Co Pte Lt","issue":"01","content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Discrete Math. Algorithm. Appl."],"published-print":{"date-parts":[[2015,3]]},"abstract":"<jats:p> Let R be a ring with the identity element 1, \u03b1 be an endomorphism of R and \u03b4 be a left \u03b1-derivation. In this paper, we introduce a generalization of a commuting graph, which is denoted by \u0393<jats:sub>R<\/jats:sub>(\u03b1, \u03b4), as a directed graph with vertex set R and, for two distinct vertices x and y, there is an arc from x to y if and only if xy = \u03b1(y)x + \u03b4(y). We study some basic properties of \u0393<jats:sub>R<\/jats:sub>(\u03b1, \u03b4). Also, we investigate the planarity and genus of the graph \u0393<jats:sub>R<\/jats:sub>(\u03b1, 0). <\/jats:p>","DOI":"10.1142\/s1793830914500682","type":"journal-article","created":{"date-parts":[[2014,9,18]],"date-time":"2014-09-18T02:52:55Z","timestamp":1411008775000},"page":"1450068","source":"Crossref","is-referenced-by-count":2,"title":["A generalization of commuting graphs"],"prefix":"10.1142","volume":"07","author":[{"given":"M.","family":"Afkhami","sequence":"first","affiliation":[{"name":"Department of Mathematics, University of Neyshabur, P. O. Box 91136-899, Neyshabur, Iran"}]},{"given":"Z.","family":"Barati","sequence":"additional","affiliation":[{"name":"Department of Mathematics, Kosar University of Bojnord, P. O. Box 9415615458, Bojnord, Iran"}]},{"given":"N.","family":"Hoseini","sequence":"additional","affiliation":[{"name":"Department of Pure Mathematics, Ferdowsi University of Mashhad, P. O. Box 1159-91775, Mashhad, Iran"}]},{"given":"K.","family":"Khashyarmanesh","sequence":"additional","affiliation":[{"name":"Department of Pure Mathematics, Ferdowsi University of Mashhad, P. O. Box 1159-91775, Mashhad, Iran"}]}],"member":"219","published-online":{"date-parts":[[2015,2,2]]},"reference":[{"key":"rf1","doi-asserted-by":"publisher","DOI":"10.1016\/j.laa.2008.01.036"},{"key":"rf2","doi-asserted-by":"publisher","DOI":"10.1016\/j.laa.2004.05.001"},{"key":"rf3","doi-asserted-by":"publisher","DOI":"10.1016\/j.laa.2006.01.029"},{"key":"rf4","doi-asserted-by":"publisher","DOI":"10.1007\/978-1-349-03521-2"},{"key":"rf5","volume-title":"Graphs and Digraphs","author":"Chartrand G.","year":"1996"},{"key":"rf6","doi-asserted-by":"publisher","DOI":"10.1016\/j.disc.2009.03.020"},{"key":"rf7","volume-title":"Algebraic Topology: An Introduction","author":"Massey W.","year":"1967"},{"key":"rf8","doi-asserted-by":"publisher","DOI":"10.1016\/0095-8956(71)90018-9"},{"key":"rf9","volume-title":"Graphs, Groups and Surfaces","author":"White A. T.","year":"1973"}],"container-title":["Discrete Mathematics, Algorithms and Applications"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.worldscientific.com\/doi\/pdf\/10.1142\/S1793830914500682","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2019,8,7]],"date-time":"2019-08-07T00:25:25Z","timestamp":1565137525000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.worldscientific.com\/doi\/abs\/10.1142\/S1793830914500682"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2015,2,2]]},"references-count":9,"journal-issue":{"issue":"01","published-online":{"date-parts":[[2015,2,2]]},"published-print":{"date-parts":[[2015,3]]}},"alternative-id":["10.1142\/S1793830914500682"],"URL":"https:\/\/doi.org\/10.1142\/s1793830914500682","relation":{},"ISSN":["1793-8309","1793-8317"],"issn-type":[{"value":"1793-8309","type":"print"},{"value":"1793-8317","type":"electronic"}],"subject":[],"published":{"date-parts":[[2015,2,2]]}}}