{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,2,21]],"date-time":"2025-02-21T10:41:59Z","timestamp":1740134519829,"version":"3.37.3"},"reference-count":20,"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":["11271006","XJEDU2012I37,I38","2012YJYB003"],"award-info":[{"award-number":["11271006","XJEDU2012I37,I38","2012YJYB003"]}],"id":[{"id":"10.13039\/501100001809","id-type":"DOI","asserted-by":"publisher"}]},{"name":"Scientific Research Programs of the Higher Education Institution of Xinjiang","award":["11271006","XJEDU2012I37,I38","2012YJYB003"],"award-info":[{"award-number":["11271006","XJEDU2012I37,I38","2012YJYB003"]}]},{"name":"Science Foundation of Changji University","award":["11271006","XJEDU2012I37,I38","2012YJYB003"],"award-info":[{"award-number":["11271006","XJEDU2012I37,I38","2012YJYB003"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Journal of Applied Mathematics"],"published-print":{"date-parts":[[2014]]},"abstract":"<jats:p>Let<mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" id=\"M1\"><mml:mrow><mml:mi>G<\/mml:mi><\/mml:mrow><\/mml:math>be a connected graph. The first and second Zagreb eccentricity indices of<mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" id=\"M2\"><mml:mrow><mml:mi>G<\/mml:mi><\/mml:mrow><\/mml:math>are defined as<mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" id=\"M3\"><mml:msubsup><mml:mrow><mml:mi>M<\/mml:mi><\/mml:mrow><mml:mrow><mml:mn mathvariant=\"normal\">1<\/mml:mn><\/mml:mrow><mml:mrow><mml:mi mathvariant=\"normal\">*<\/mml:mi><\/mml:mrow><\/mml:msubsup><mml:mo stretchy=\"false\">(<\/mml:mo><mml:mi>G<\/mml:mi><mml:mo stretchy=\"false\">)<\/mml:mo><mml:mo>=<\/mml:mo><mml:mrow><mml:msub><mml:mo stretchy=\"false\">\u2211<\/mml:mo><mml:mrow><mml:mi>v<\/mml:mi><mml:mo>\u2208<\/mml:mo><mml:mi>V<\/mml:mi><mml:mo stretchy=\"false\">(<\/mml:mo><mml:mi>G<\/mml:mi><mml:mo stretchy=\"false\">)<\/mml:mo><\/mml:mrow><\/mml:msub><mml:mrow><mml:mi mathvariant=\"normal\">\u200d<\/mml:mi><\/mml:mrow><\/mml:mrow><mml:msubsup><mml:mrow><mml:mi>\u03b5<\/mml:mi><\/mml:mrow><mml:mrow><mml:mi>G<\/mml:mi><\/mml:mrow><mml:mrow><mml:mn mathvariant=\"normal\">2<\/mml:mn><\/mml:mrow><\/mml:msubsup><mml:mo stretchy=\"false\">(<\/mml:mo><mml:mi>v<\/mml:mi><mml:mo stretchy=\"false\">)<\/mml:mo><\/mml:math>and<mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" id=\"M4\"><mml:msubsup><mml:mrow><mml:mi>M<\/mml:mi><\/mml:mrow><mml:mrow><mml:mn mathvariant=\"normal\">2<\/mml:mn><\/mml:mrow><mml:mrow><mml:mi mathvariant=\"normal\">*<\/mml:mi><\/mml:mrow><\/mml:msubsup><mml:mo stretchy=\"false\">(<\/mml:mo><mml:mi>G<\/mml:mi><mml:mo stretchy=\"false\">)<\/mml:mo><mml:mo>=<\/mml:mo><mml:mrow><mml:msub><mml:mo stretchy=\"false\">\u2211<\/mml:mo><mml:mrow><mml:mi>u<\/mml:mi><mml:mi>v<\/mml:mi><mml:mo>\u2208<\/mml:mo><mml:mi>E<\/mml:mi><mml:mo stretchy=\"false\">(<\/mml:mo><mml:mi>G<\/mml:mi><mml:mo stretchy=\"false\">)<\/mml:mo><\/mml:mrow><\/mml:msub><mml:mrow><mml:mi mathvariant=\"normal\">\u200d<\/mml:mi><\/mml:mrow><\/mml:mrow><mml:msub><mml:mrow><mml:mi>\u03b5<\/mml:mi><\/mml:mrow><mml:mrow><mml:mi>G<\/mml:mi><\/mml:mrow><\/mml:msub><mml:mo stretchy=\"false\">(<\/mml:mo><mml:mi>u<\/mml:mi><mml:mo stretchy=\"false\">)<\/mml:mo><mml:msub><mml:mrow><mml:mi>\u03b5<\/mml:mi><\/mml:mrow><mml:mrow><mml:mi>G<\/mml:mi><\/mml:mrow><\/mml:msub><mml:mo stretchy=\"false\">(<\/mml:mo><mml:mi>v<\/mml:mi><mml:mo stretchy=\"false\">)<\/mml:mo><\/mml:math>, where<mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" id=\"M5\"><mml:msub><mml:mrow><mml:mi>\u03b5<\/mml:mi><\/mml:mrow><mml:mrow><mml:mi>G<\/mml:mi><\/mml:mrow><\/mml:msub><mml:mo stretchy=\"false\">(<\/mml:mo><mml:mi>v<\/mml:mi><mml:mo stretchy=\"false\">)<\/mml:mo><\/mml:math>is the eccentricity of the vertex<mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" id=\"M6\"><mml:mrow><mml:mi>v<\/mml:mi><\/mml:mrow><\/mml:math>in<mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" id=\"M7\"><mml:mrow><mml:mi>G<\/mml:mi><\/mml:mrow><\/mml:math>and<mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" id=\"M8\"><mml:msubsup><mml:mrow><mml:mi>\u03b5<\/mml:mi><\/mml:mrow><mml:mrow><mml:mi>G<\/mml:mi><\/mml:mrow><mml:mrow><mml:mn mathvariant=\"normal\">2<\/mml:mn><\/mml:mrow><\/mml:msubsup><mml:mo stretchy=\"false\">(<\/mml:mo><mml:mi>v<\/mml:mi><mml:mo stretchy=\"false\">)<\/mml:mo><mml:mo>=<\/mml:mo><mml:mo stretchy=\"false\">(<\/mml:mo><mml:msub><mml:mrow><mml:mi>\u03b5<\/mml:mi><\/mml:mrow><mml:mrow><mml:mi>G<\/mml:mi><\/mml:mrow><\/mml:msub><mml:mo stretchy=\"false\">(<\/mml:mo><mml:mi>v<\/mml:mi><mml:mo stretchy=\"false\">)<\/mml:mo><mml:msup><mml:mrow><mml:mo stretchy=\"false\">)<\/mml:mo><\/mml:mrow><mml:mrow><mml:mn mathvariant=\"normal\">2<\/mml:mn><\/mml:mrow><\/mml:msup><\/mml:math>. Suppose that<mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" id=\"M9\"><mml:mi>G<\/mml:mi><mml:mo stretchy=\"false\">(<\/mml:mo><mml:mi>U<\/mml:mi><mml:mo stretchy=\"false\">)<\/mml:mo><mml:mo>\u2293<\/mml:mo><mml:mi>H<\/mml:mi><mml:mo stretchy=\"false\">(<\/mml:mo><mml:mi mathvariant=\"normal\">\u2205<\/mml:mi><mml:mo>\u2260<\/mml:mo><mml:mi>U<\/mml:mi><mml:mo>\u2286<\/mml:mo><mml:mi>V<\/mml:mi><mml:mo stretchy=\"false\">(<\/mml:mo><mml:mi>G<\/mml:mi><mml:mo stretchy=\"false\">)<\/mml:mo><mml:mo stretchy=\"false\">)<\/mml:mo><\/mml:math>is the generalized hierarchical product of two connected graphs<mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" id=\"M10\"><mml:mrow><mml:mi>G<\/mml:mi><\/mml:mrow><\/mml:math>and<mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" id=\"M11\"><mml:mrow><mml:mi>H<\/mml:mi><\/mml:mrow><\/mml:math>. In this paper, the Zagreb eccentricity indices<mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" id=\"M12\"><mml:mrow><mml:msubsup><mml:mrow><mml:mi>M<\/mml:mi><\/mml:mrow><mml:mrow><mml:mn mathvariant=\"normal\">1<\/mml:mn><\/mml:mrow><mml:mrow><mml:mi mathvariant=\"normal\">*<\/mml:mi><\/mml:mrow><\/mml:msubsup><\/mml:mrow><\/mml:math>and<mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" id=\"M13\"><mml:mrow><mml:msubsup><mml:mrow><mml:mi>M<\/mml:mi><\/mml:mrow><mml:mrow><mml:mn mathvariant=\"normal\">2<\/mml:mn><\/mml:mrow><mml:mrow><mml:mi mathvariant=\"normal\">*<\/mml:mi><\/mml:mrow><\/mml:msubsup><\/mml:mrow><\/mml:math>of<mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" id=\"M14\"><mml:mi>G<\/mml:mi><mml:mo stretchy=\"false\">(<\/mml:mo><mml:mi>U<\/mml:mi><mml:mo stretchy=\"false\">)<\/mml:mo><mml:mo>\u2293<\/mml:mo><mml:mi>H<\/mml:mi><\/mml:math>are computed. Moreover, we present explicit formulas for the<mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" id=\"M15\"><mml:mrow><mml:msubsup><mml:mrow><mml:mi>M<\/mml:mi><\/mml:mrow><mml:mrow><mml:mn mathvariant=\"normal\">1<\/mml:mn><\/mml:mrow><mml:mrow><mml:mi mathvariant=\"normal\">*<\/mml:mi><\/mml:mrow><\/mml:msubsup><\/mml:mrow><\/mml:math>and<mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" id=\"M16\"><mml:mrow><mml:msubsup><mml:mrow><mml:mi>M<\/mml:mi><\/mml:mrow><mml:mrow><mml:mn mathvariant=\"normal\">2<\/mml:mn><\/mml:mrow><mml:mrow><mml:mi mathvariant=\"normal\">*<\/mml:mi><\/mml:mrow><\/mml:msubsup><\/mml:mrow><\/mml:math>of<jats:italic>S<\/jats:italic>-sum graph, Cartesian, cluster, and corona product graphs by means of some invariants of the factors.<\/jats:p>","DOI":"10.1155\/2014\/241712","type":"journal-article","created":{"date-parts":[[2014,2,10]],"date-time":"2014-02-10T16:01:17Z","timestamp":1392048077000},"page":"1-8","source":"Crossref","is-referenced-by-count":7,"title":["Zagreb Eccentricity Indices of the Generalized Hierarchical Product Graphs and Their Applications"],"prefix":"10.1155","volume":"2014","author":[{"given":"Zhaoyang","family":"Luo","sequence":"first","affiliation":[{"name":"School of Mathematics, Shandong University, Jinan 250100, China"},{"name":"Department of Mathematics, Changji University, Changji 831100, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Jianliang","family":"Wu","sequence":"additional","affiliation":[{"name":"School of Mathematics, Shandong University, Jinan 250100, China"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"311","reference":[{"key":"11","doi-asserted-by":"publisher","DOI":"10.1007\/978-3-642-70982-1"},{"issue":"1","key":"18","doi-asserted-by":"crossref","first-page":"17","DOI":"10.1021\/ja01193a005","volume":"69","year":"1947","journal-title":"Journal of the American Chemical Society"},{"key":"12","doi-asserted-by":"publisher","DOI":"10.1016\/0009-2614(72)85099-1"},{"key":"10","doi-asserted-by":"publisher","DOI":"10.2298\/FIL1201093G"},{"issue":"3","key":"16","first-page":"524","volume":"57","year":"2010","journal-title":"Acta Chimica Slovenica"},{"issue":"1","key":"5","first-page":"117","volume":"6","year":"2013","journal-title":"Ars Mathematica Contemporanea"},{"issue":"3","key":"6","doi-asserted-by":"crossref","first-page":"359","DOI":"10.5562\/cca2020","volume":"85","year":"2012","journal-title":"Croatica Chemica Acta"},{"key":"19","doi-asserted-by":"publisher","DOI":"10.5562\/cca1801"},{"key":"3","doi-asserted-by":"publisher","DOI":"10.1016\/j.dam.2008.04.018"},{"key":"4","doi-asserted-by":"publisher","DOI":"10.1016\/j.disc.2008.10.028"},{"issue":"3","key":"1","first-page":"591","volume":"64","year":"2010","journal-title":"MATCH. 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