{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,3,8]],"date-time":"2026-03-08T07:03:47Z","timestamp":1772953427938,"version":"3.50.1"},"reference-count":30,"publisher":"Wiley","license":[{"start":{"date-parts":[[2020,5,31]],"date-time":"2020-05-31T00:00:00Z","timestamp":1590883200000},"content-version":"unspecified","delay-in-days":0,"URL":"http:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"DOI":"10.13039\/501100006013","name":"United Arab Emirates University","doi-asserted-by":"publisher","award":["G00002590"],"award-info":[{"award-number":["G00002590"]}],"id":[{"id":"10.13039\/501100006013","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100006013","name":"United Arab Emirates University","doi-asserted-by":"publisher","award":["G00003271"],"award-info":[{"award-number":["G00003271"]}],"id":[{"id":"10.13039\/501100006013","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100006013","name":"United Arab Emirates University","doi-asserted-by":"publisher","award":["G00002590"],"award-info":[{"award-number":["G00002590"]}],"id":[{"id":"10.13039\/501100006013","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100006013","name":"United Arab Emirates University","doi-asserted-by":"publisher","award":["G00003271"],"award-info":[{"award-number":["G00003271"]}],"id":[{"id":"10.13039\/501100006013","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Complexity"],"published-print":{"date-parts":[[2020,5,31]]},"abstract":"<jats:p>The eccentric connectivity polynomial (ECP) of a connected graph<mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" id=\"M2\"><mml:mi>G<\/mml:mi><mml:mo>=<\/mml:mo><mml:mfenced open=\"(\" close=\")\" separators=\"|\"><mml:mrow><mml:mi>V<\/mml:mi><mml:mfenced open=\"(\" close=\")\" separators=\"|\"><mml:mrow><mml:mi>G<\/mml:mi><\/mml:mrow><\/mml:mfenced><mml:mo>,<\/mml:mo><mml:mi>E<\/mml:mi><mml:mfenced open=\"(\" close=\")\" separators=\"|\"><mml:mrow><mml:mi>G<\/mml:mi><\/mml:mrow><\/mml:mfenced><\/mml:mrow><\/mml:mfenced><\/mml:math>is described as<mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" id=\"M3\"><mml:msup><mml:mrow><mml:mi>\u03be<\/mml:mi><\/mml:mrow><mml:mrow><mml:mi>c<\/mml:mi><\/mml:mrow><\/mml:msup><mml:mfenced open=\"(\" close=\")\" separators=\"|\"><mml:mrow><mml:mi>G<\/mml:mi><mml:mo>,<\/mml:mo><mml:mi>y<\/mml:mi><\/mml:mrow><\/mml:mfenced><mml:mo>=<\/mml:mo><mml:mstyle displaystyle=\"true\"><mml:msub><mml:mrow><mml:mo stretchy=\"false\">\u2211<\/mml:mo><\/mml:mrow><mml:mrow><mml:mi>a<\/mml:mi><mml:mo>\u2208<\/mml:mo><mml:mi>V<\/mml:mi><mml:mfenced open=\"(\" close=\")\" separators=\"|\"><mml:mrow><mml:mi>G<\/mml:mi><\/mml:mrow><\/mml:mfenced><\/mml:mrow><\/mml:msub><mml:mrow><mml:msub><mml:mrow><mml:mi mathvariant=\"normal\">deg<\/mml:mi><\/mml:mrow><mml:mrow><mml:mi>G<\/mml:mi><\/mml:mrow><\/mml:msub><mml:mfenced open=\"(\" close=\")\" separators=\"|\"><mml:mrow><mml:mi>a<\/mml:mi><\/mml:mrow><\/mml:mfenced><mml:msup><mml:mrow><mml:mi>y<\/mml:mi><\/mml:mrow><mml:mrow><mml:mi>e<\/mml:mi><mml:msub><mml:mrow><mml:mi>c<\/mml:mi><\/mml:mrow><mml:mrow><mml:mi>G<\/mml:mi><\/mml:mrow><\/mml:msub><mml:mfenced open=\"(\" close=\")\" separators=\"|\"><mml:mrow><mml:mi>a<\/mml:mi><\/mml:mrow><\/mml:mfenced><\/mml:mrow><\/mml:msup><\/mml:mrow><\/mml:mstyle><\/mml:math>, where<mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" id=\"M4\"><mml:mi>e<\/mml:mi><mml:msub><mml:mrow><mml:mi>c<\/mml:mi><\/mml:mrow><mml:mrow><mml:mi>G<\/mml:mi><\/mml:mrow><\/mml:msub><mml:mfenced open=\"(\" close=\")\" separators=\"|\"><mml:mrow><mml:mi>a<\/mml:mi><\/mml:mrow><\/mml:mfenced><\/mml:math>and<mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" id=\"M5\"><mml:msub><mml:mrow><mml:mi mathvariant=\"normal\">deg<\/mml:mi><\/mml:mrow><mml:mrow><mml:mi>G<\/mml:mi><\/mml:mrow><\/mml:msub><mml:mfenced open=\"(\" close=\")\" separators=\"|\"><mml:mrow><mml:mi>a<\/mml:mi><\/mml:mrow><\/mml:mfenced><\/mml:math>represent the eccentricity and the degree of the vertex<mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" id=\"M6\"><mml:mi>a<\/mml:mi><\/mml:math>, respectively. The eccentric connectivity index (ECI) can also be acquired from<mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" id=\"M7\"><mml:msup><mml:mrow><mml:mi>\u03be<\/mml:mi><\/mml:mrow><mml:mrow><mml:mi>c<\/mml:mi><\/mml:mrow><\/mml:msup><mml:mfenced open=\"(\" close=\")\" separators=\"|\"><mml:mrow><mml:mi>G<\/mml:mi><mml:mo>,<\/mml:mo><mml:mi>y<\/mml:mi><\/mml:mrow><\/mml:mfenced><\/mml:math>by taking its first derivatives at<mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" id=\"M8\"><mml:mi>y<\/mml:mi><mml:mo>=<\/mml:mo><mml:mn>1<\/mml:mn><\/mml:math>. The ECI has been widely used for analyzing both the boiling point and melting point for chemical compounds and medicinal drugs in QSPR\/QSAR studies. As the extension of ECI, the ECP also performs a pivotal role in pharmaceutical science and chemical engineering. Graph products conveniently play an important role in many combinatorial applications, graph decompositions, pure mathematics, and applied mathematics. In this article, we work out the ECP of<mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" id=\"M9\"><mml:mi mathvariant=\"normal\">\u2131<\/mml:mi><\/mml:math>-sum of graphs. Moreover, we derive the explicit expressions of ECP for well-known graph products such as generalized hierarchical, cluster, and corona products of graphs. We also apply these outcomes to deduce the ECP of some classes of chemical graphs.<\/jats:p>","DOI":"10.1155\/2020\/5061682","type":"journal-article","created":{"date-parts":[[2020,5,31]],"date-time":"2020-05-31T23:30:55Z","timestamp":1590967855000},"page":"1-9","source":"Crossref","is-referenced-by-count":12,"title":["On the Eccentric Connectivity Polynomial of<mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" id=\"M1\"><mml:mi mathvariant=\"normal\">\u2131<\/mml:mi><\/mml:math>-Sum of Connected Graphs"],"prefix":"10.1155","volume":"2020","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-2827-0462","authenticated-orcid":true,"given":"Muhammad","family":"Imran","sequence":"first","affiliation":[{"name":"Department of Mathematical Sciences, College of Science, United Arab Emirates University, P.O. Box 15551, Al Ain, UAE"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Shehnaz","family":"Akhter","sequence":"additional","affiliation":[{"name":"Department of Mathematics, School of Natural Sciences, National University of Sciences and Technology, Sector H-12, Islamabad, Pakistan"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-1549-3584","authenticated-orcid":true,"given":"Zahid","family":"Iqbal","sequence":"additional","affiliation":[{"name":"Department of Mathematics, School of Natural Sciences, National University of Sciences and Technology, Sector H-12, Islamabad, Pakistan"},{"name":"Department of Mathematics and Statistics, Institute of Southern Punjab, Multan, Pakistan"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"311","reference":[{"key":"1","doi-asserted-by":"publisher","DOI":"10.1021\/ci960049h"},{"key":"2","doi-asserted-by":"publisher","DOI":"10.1016\/j.ejc.2010.03.007"},{"key":"3","doi-asserted-by":"publisher","DOI":"10.1016\/j.dam.2017.02.009"},{"key":"4","doi-asserted-by":"publisher","DOI":"10.1016\/j.ejc.2012.05.005"},{"key":"5","doi-asserted-by":"publisher","DOI":"10.1155\/2020\/6079507"},{"key":"6","doi-asserted-by":"publisher","DOI":"10.1142\/s1793830918500775"},{"key":"7","doi-asserted-by":"publisher","DOI":"10.3390\/sym10070237"},{"issue":"8","key":"8","first-page":"823","volume":"3","year":"2009","journal-title":"Optoelectronics and Advanced Materials\u2014Rapid Communications"},{"key":"9","doi-asserted-by":"publisher","DOI":"10.1186\/s13660-016-1186-x"},{"key":"10","doi-asserted-by":"publisher","DOI":"10.1016\/j.akcej.2016.11.012"},{"key":"11","doi-asserted-by":"publisher","DOI":"10.1007\/s40819-014-0016-4"},{"issue":"1","key":"12","first-page":"103","volume":"2","year":"2013","journal-title":"Transactions on Combinatorics"},{"key":"13","first-page":"3","volume":"95","year":"2014","journal-title":"Utilitas Mathematica"},{"key":"14","first-page":"297","volume":"84","year":"2011","journal-title":"Utilitas Mathematica"},{"issue":"3","key":"15","first-page":"483","volume":"4","year":"2009","journal-title":"Digest Journal of Nanomaterials and Biostructures"},{"issue":"7","key":"16","first-page":"761","volume":"5","year":"2011","journal-title":"Optoelectronics and Advanced Materials\u2014Rapid Communications"},{"issue":"3","key":"17","first-page":"545","volume":"4","year":"2009","journal-title":"Digest Journal of Nanomaterials and Biostructures"},{"issue":"1","key":"18","first-page":"5","volume":"45","year":"2013","journal-title":"Bulgarian Chemical Communications"},{"issue":"12","key":"19","first-page":"1306","volume":"3","year":"2009","journal-title":"Optoelectronics and Advanced Materials\u2014Rapid Communications"},{"key":"20","doi-asserted-by":"publisher","DOI":"10.1080\/1536383x.2011.588810"},{"key":"21","doi-asserted-by":"publisher","DOI":"10.1166\/jctn.2014.3378"},{"key":"22","doi-asserted-by":"publisher","DOI":"10.1109\/access.2019.2944860"},{"key":"23","doi-asserted-by":"crossref","first-page":"101","DOI":"10.55630\/sjc.2011.5.101-116","volume":"5","year":"2011","journal-title":"Serdica Journal of Computing"},{"key":"24","doi-asserted-by":"publisher","DOI":"10.1016\/j.disc.2008.10.028"},{"key":"25","doi-asserted-by":"publisher","DOI":"10.1016\/j.dam.2008.04.018"},{"key":"26","doi-asserted-by":"publisher","DOI":"10.1016\/j.dam.2010.12.020"},{"key":"27","doi-asserted-by":"publisher","DOI":"10.1016\/j.dam.2008.07.001"},{"key":"28","doi-asserted-by":"publisher","DOI":"10.1016\/j.dam.2011.06.001"},{"key":"29","doi-asserted-by":"publisher","DOI":"10.1016\/j.dam.2010.04.003"},{"key":"30","first-page":"745","volume":"65","year":"2011","journal-title":"MATCH Communications in Mathematical and in Computer Chemistry"}],"container-title":["Complexity"],"original-title":[],"language":"en","link":[{"URL":"http:\/\/downloads.hindawi.com\/journals\/complexity\/2020\/5061682.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"http:\/\/downloads.hindawi.com\/journals\/complexity\/2020\/5061682.xml","content-type":"application\/xml","content-version":"vor","intended-application":"text-mining"},{"URL":"http:\/\/downloads.hindawi.com\/journals\/complexity\/2020\/5061682.pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2023,10,1]],"date-time":"2023-10-01T19:23:04Z","timestamp":1696188184000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.hindawi.com\/journals\/complexity\/2020\/5061682\/"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2020,5,31]]},"references-count":30,"alternative-id":["5061682","5061682"],"URL":"https:\/\/doi.org\/10.1155\/2020\/5061682","relation":{},"ISSN":["1076-2787","1099-0526"],"issn-type":[{"value":"1076-2787","type":"print"},{"value":"1099-0526","type":"electronic"}],"subject":[],"published":{"date-parts":[[2020,5,31]]}}}