{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,3,17]],"date-time":"2026-03-17T13:32:06Z","timestamp":1773754326422,"version":"3.50.1"},"reference-count":32,"publisher":"Wiley","license":[{"start":{"date-parts":[[2014,1,1]],"date-time":"2014-01-01T00:00:00Z","timestamp":1388534400000},"content-version":"unspecified","delay-in-days":0,"URL":"http:\/\/creativecommons.org\/licenses\/by\/3.0\/"}],"funder":[{"DOI":"10.13039\/501100001809","name":"National Natural Science Foundation of China","doi-asserted-by":"publisher","award":["11371180"],"award-info":[{"award-number":["11371180"]}],"id":[{"id":"10.13039\/501100001809","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100001809","name":"National Natural Science Foundation of China","doi-asserted-by":"publisher","award":["lzujbky-2014-21"],"award-info":[{"award-number":["lzujbky-2014-21"]}],"id":[{"id":"10.13039\/501100001809","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100012226","name":"Fundamental Research Funds for the Central Universities","doi-asserted-by":"crossref","award":["11371180"],"award-info":[{"award-number":["11371180"]}],"id":[{"id":"10.13039\/501100012226","id-type":"DOI","asserted-by":"crossref"}]},{"DOI":"10.13039\/501100012226","name":"Fundamental Research Funds for the Central Universities","doi-asserted-by":"crossref","award":["lzujbky-2014-21"],"award-info":[{"award-number":["lzujbky-2014-21"]}],"id":[{"id":"10.13039\/501100012226","id-type":"DOI","asserted-by":"crossref"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Journal of Applied Mathematics"],"published-print":{"date-parts":[[2014]]},"abstract":"Fowler and Pisanski showed that the Fries number for a fullerene on surface \u03a3 is bounded above by|<\/mml:mo>V<\/mml:mi>|<\/mml:mo>\/<\/mml:mo>3<\/mml:mn><\/mml:math>, and fullerenes which attain this bound are exactly the class of leapfrog fullerenes on surface \u03a3. We showed that the Clar number of a fullerene on surface \u03a3 is bounded above by(<\/mml:mo>|<\/mml:mo>V<\/mml:mi>|<\/mml:mo>\/<\/mml:mo>6<\/mml:mn>)<\/mml:mo>-<\/mml:mo>\u03c7<\/mml:mi>(<\/mml:mo>\u03a3<\/mml:mi>)<\/mml:mo><\/mml:math>, where\u03c7<\/mml:mi>(<\/mml:mo>\u03a3<\/mml:mi>)<\/mml:mo><\/mml:math>stands for the Euler characteristic of \u03a3. By establishing a relation between the extremal fullerenes and the extremal (4,6)-fullerenes on the sphere, Hartung characterized the fullerenes on the sphereS<\/mml:mi><\/mml:mrow>0<\/mml:mn><\/mml:mrow><\/mml:msub><\/mml:mrow><\/mml:math>for which Clar numbers attain(<\/mml:mo>|<\/mml:mo>V<\/mml:mi>|<\/mml:mo>\/<\/mml:mo>6<\/mml:mn>)<\/mml:mo>-<\/mml:mo>\u03c7<\/mml:mi>(<\/mml:mo>S<\/mml:mi><\/mml:mrow>0<\/mml:mn><\/mml:mrow><\/mml:msub>)<\/mml:mo><\/mml:math>. We prove that, for a (4,6)-fullerene on surface \u03a3, its Clar number is bounded above by(<\/mml:mo>|<\/mml:mo>V<\/mml:mi>|<\/mml:mo>\/<\/mml:mo>6<\/mml:mn>)<\/mml:mo>+<\/mml:mo>\u03c7<\/mml:mi>(<\/mml:mo>\u03a3<\/mml:mi>)<\/mml:mo><\/mml:math>and its Fries number is bounded above by(<\/mml:mo>|<\/mml:mo>V<\/mml:mi>|<\/mml:mo>\/<\/mml:mo>3<\/mml:mn>)<\/mml:mo>+<\/mml:mo>\u03c7<\/mml:mi>(<\/mml:mo>\u03a3<\/mml:mi>)<\/mml:mo><\/mml:math>, and we characterize the (4,6)-fullerenes on surface \u03a3 attaining these two bounds in terms of perfect Clar structure. Moreover, we characterize the fullerenes on the projective planeN<\/mml:mi><\/mml:mrow>1<\/mml:mn><\/mml:mrow><\/mml:msub><\/mml:mrow><\/mml:math>for which Clar numbers attain(<\/mml:mo>|<\/mml:mo>V<\/mml:mi>|<\/mml:mo>\/<\/mml:mo>6<\/mml:mn>)<\/mml:mo>-<\/mml:mo>\u03c7<\/mml:mi>(<\/mml:mo>N<\/mml:mi><\/mml:mrow>1<\/mml:mn><\/mml:mrow><\/mml:msub>)<\/mml:mo><\/mml:math>in Hartung\u2019s method.<\/jats:p>","DOI":"10.1155\/2014\/196792","type":"journal-article","created":{"date-parts":[[2014,8,5]],"date-time":"2014-08-05T21:02:00Z","timestamp":1407272520000},"page":"1-11","source":"Crossref","is-referenced-by-count":6,"title":["Clar Structure and Fries Set of Fullerenes and (4,6)-Fullerenes on Surfaces"],"prefix":"10.1155","volume":"2014","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-5690-7408","authenticated-orcid":true,"given":"Yang","family":"Gao","sequence":"first","affiliation":[{"name":"School of Mathematics and Statistics, Lanzhou University, Lanzhou, Gansu 730000, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Heping","family":"Zhang","sequence":"additional","affiliation":[{"name":"School of Mathematics and Statistics, Lanzhou University, Lanzhou, Gansu 730000, China"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"311","reference":[{"key":"2","first-page":"87","volume":"33","year":"1996","journal-title":"MATCH: Communications in Mathematical and in Computer Chemistry"},{"key":"6","doi-asserted-by":"publisher","DOI":"10.1017\/CBO9780511721311"},{"key":"11","doi-asserted-by":"publisher","DOI":"10.1039\/ft9969202197"},{"key":"21","doi-asserted-by":"publisher","DOI":"10.1016\/j.dam.2011.05.006"},{"issue":"3","key":"29","first-page":"643","volume":"57","year":"2007","journal-title":"MATCH Communications in Mathematical and in Computer Chemistry"},{"key":"12","series-title":"DIMACS Series","first-page":"139","volume-title":"(3, 6)-cages, hexagonal torodial cages, and their 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