{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,2,21]],"date-time":"2025-02-21T14:26:01Z","timestamp":1740147961568,"version":"3.37.3"},"reference-count":9,"publisher":"Wiley","license":[{"start":{"date-parts":[[2016,1,1]],"date-time":"2016-01-01T00:00:00Z","timestamp":1451606400000},"content-version":"unspecified","delay-in-days":0,"URL":"http:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"name":"NSF of the Higher Education Institutions of Jiangsu Province","award":["12KJB110001","11571135"],"award-info":[{"award-number":["12KJB110001","11571135"]}]},{"DOI":"10.13039\/501100001809","name":"National Natural Science Foundation of China","doi-asserted-by":"publisher","award":["12KJB110001","11571135"],"award-info":[{"award-number":["12KJB110001","11571135"]}],"id":[{"id":"10.13039\/501100001809","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Computational and Mathematical Methods in Medicine"],"published-print":{"date-parts":[[2016]]},"abstract":"<jats:p>Graphs are used to model chemical compounds and drugs. In the graphs, each vertex represents an atom of molecule and edges between the corresponding vertices are used to represent covalent bounds between atoms. The<jats:italic>Wiener polarity index<\/jats:italic><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" id=\"M1\"><mml:msub><mml:mrow><mml:mi>W<\/mml:mi><\/mml:mrow><mml:mrow><mml:mi>p<\/mml:mi><\/mml:mrow><\/mml:msub><mml:mo stretchy=\"false\">(<\/mml:mo><mml:mi>G<\/mml:mi><mml:mo stretchy=\"false\">)<\/mml:mo><\/mml:math>of a graph<mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" id=\"M2\"><mml:mrow><mml:mi>G<\/mml:mi><\/mml:mrow><\/mml:math>is the number of unordered pairs of vertices<mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" id=\"M3\"><mml:mi>u<\/mml:mi><mml:mo>,<\/mml:mo><mml:mi>v<\/mml:mi><\/mml:math>of<mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" id=\"M4\"><mml:mrow><mml:mi>G<\/mml:mi><\/mml:mrow><\/mml:math>such that the distance between<mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" id=\"M5\"><mml:mrow><mml:mi>u<\/mml:mi><\/mml:mrow><\/mml:math>and<mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" id=\"M6\"><mml:mrow><mml:mi>v<\/mml:mi><\/mml:mrow><\/mml:math>is equal to 3. The trees and unicyclic graphs with perfect matching, of which all vertices have degrees not greater than three, are referred to as the<jats:italic>H\u00fcckel trees<\/jats:italic>and<jats:italic>unicyclic H\u00fcckel graphs<\/jats:italic>, respectively. In this paper, we first consider the smallest and the largest Wiener polarity index among all H\u00fcckel trees on<mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" id=\"M7\"><mml:mn mathvariant=\"normal\">2<\/mml:mn><mml:mi>n<\/mml:mi><\/mml:math>vertices and characterize the corresponding extremal graphs. Then we obtain an upper and lower bound for the Wiener polarity index of unicyclic H\u00fcckel graphs on<mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" id=\"M8\"><mml:mn mathvariant=\"normal\">2<\/mml:mn><mml:mi>n<\/mml:mi><\/mml:math>vertices.<\/jats:p>","DOI":"10.1155\/2016\/3873597","type":"journal-article","created":{"date-parts":[[2016,5,9]],"date-time":"2016-05-09T17:03:16Z","timestamp":1462813396000},"page":"1-6","source":"Crossref","is-referenced-by-count":1,"title":["On the Extremal Wiener Polarity Index of H\u00fcckel Graphs"],"prefix":"10.1155","volume":"2016","author":[{"given":"Hongzhuan","family":"Wang","sequence":"first","affiliation":[{"name":"Faculty of Mathematics and Physics, Huaiyin Institute of Technology, Huai\u2019an, Jiangsu 223003, China"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"311","reference":[{"key":"9","doi-asserted-by":"publisher","DOI":"10.1021\/ja01193a005"},{"key":"8","doi-asserted-by":"publisher","DOI":"10.1021\/ci970122j"},{"volume-title":"Mathematical and chemical analysis of Wiener's polarity number","year":"2002","key":"4"},{"issue":"1","key":"1","first-page":"235","volume":"62","year":"2009","journal-title":"MATCH Communications In Mathematical and in Computer Chemistry"},{"key":"2","first-page":"257","volume":"63","year":"2010","journal-title":"MATCH Communications in Mathematical and in Computer Chemistry"},{"issue":"1","key":"3","first-page":"305","volume":"66","year":"2011","journal-title":"MATCH Communications in Mathematical and in Computer Chemistry"},{"key":"5","doi-asserted-by":"publisher","DOI":"10.1016\/j.amc.2012.03.090"},{"key":"6","doi-asserted-by":"publisher","DOI":"10.1016\/s0024-3795(01)00465-7"},{"issue":"1","key":"7","first-page":"293","volume":"66","year":"2011","journal-title":"MATCH Communications in Mathematical and in Computer Chemistry"}],"container-title":["Computational and Mathematical Methods in Medicine"],"original-title":[],"language":"en","link":[{"URL":"http:\/\/downloads.hindawi.com\/journals\/cmmm\/2016\/3873597.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"http:\/\/downloads.hindawi.com\/journals\/cmmm\/2016\/3873597.xml","content-type":"application\/xml","content-version":"vor","intended-application":"text-mining"},{"URL":"http:\/\/downloads.hindawi.com\/journals\/cmmm\/2016\/3873597.pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2016,8,18]],"date-time":"2016-08-18T14:00:45Z","timestamp":1471528845000},"score":1,"resource":{"primary":{"URL":"http:\/\/www.hindawi.com\/journals\/cmmm\/2016\/3873597\/"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2016]]},"references-count":9,"alternative-id":["3873597","3873597"],"URL":"https:\/\/doi.org\/10.1155\/2016\/3873597","relation":{},"ISSN":["1748-670X","1748-6718"],"issn-type":[{"type":"print","value":"1748-670X"},{"type":"electronic","value":"1748-6718"}],"subject":[],"published":{"date-parts":[[2016]]}}}