{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2023,10,24]],"date-time":"2023-10-24T00:16:15Z","timestamp":1698106575138},"reference-count":11,"publisher":"Wiley","issue":"1","license":[{"start":{"date-parts":[[2006,10,11]],"date-time":"2006-10-11T00:00:00Z","timestamp":1160524800000},"content-version":"vor","delay-in-days":5762,"URL":"http:\/\/onlinelibrary.wiley.com\/termsAndConditions#vor"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Networks"],"published-print":{"date-parts":[[1991,1]]},"abstract":"<jats:title>Abstract<\/jats:title><jats:p>If <jats:italic>k<\/jats:italic> is a nonnegative integer and <jats:italic>x<\/jats:italic> is a vertex of a tree <jats:italic>T<\/jats:italic>, the <jats:italic>k<\/jats:italic>\u2010ball branch weight of <jats:italic>x<\/jats:italic>, denoted <jats:italic>b(x;k)<\/jats:italic>, is the number of vertices in a largest subtree of <jats:italic>T<\/jats:italic>, all of whose vertices are a distance at least <jats:italic>k<\/jats:italic> + 1 from <jats:italic>x<\/jats:italic>. The <jats:italic>k<\/jats:italic>\u2010ball branch weight centroid of <jats:italic>T<\/jats:italic>, denoted <jats:italic>W(T;k)<\/jats:italic>, consists of all vertices <jats:italic>x<\/jats:italic> of <jats:italic>T<\/jats:italic> for which <jats:italic>b(x;k)<\/jats:italic> is a minimum. The usual branch weight centroid of <jats:italic>T<\/jats:italic> (which is also the median) is <jats:italic>W<\/jats:italic>(<jats:italic>T<\/jats:italic>;0), and the center of <jats:italic>T<\/jats:italic> is <jats:italic>W(T;r)<\/jats:italic>, where <jats:italic>r<\/jats:italic> is the radius of <jats:italic>T<\/jats:italic>. In this paper, the structure of the subgraph spanned by <jats:italic>W(T;k)<\/jats:italic> is examined. Similarities and differences with Slater's <jats:italic>k<\/jats:italic>\u2010centrum [6] and <jats:italic>k<\/jats:italic>\u2010nucleus [7] are discussed.<\/jats:p>","DOI":"10.1002\/net.3230210103","type":"journal-article","created":{"date-parts":[[2007,5,12]],"date-time":"2007-05-12T10:17:58Z","timestamp":1178965078000},"page":"11-17","source":"Crossref","is-referenced-by-count":9,"title":["Centroids to centers in trees"],"prefix":"10.1002","volume":"21","author":[{"given":"K. B.","family":"Reid","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"311","published-online":{"date-parts":[[2006,10,11]]},"reference":[{"key":"e_1_2_1_2_2","first-page":"285","article-title":"The centrality of vertices in trees","volume":"9","author":"\u00c1d\u00e1m A.","year":"1974","journal-title":"Stud. Sci. Math. Hung."},{"key":"e_1_2_1_3_2","volume-title":"Location on Networks","author":"Handler G. Y.","year":"1979"},{"key":"e_1_2_1_4_2","doi-asserted-by":"publisher","DOI":"10.1016\/0012-365X(71)90003-3"},{"key":"e_1_2_1_5_2","doi-asserted-by":"publisher","DOI":"10.1515\/crll.1869.70.185"},{"key":"e_1_2_1_6_2","doi-asserted-by":"publisher","DOI":"10.1007\/BF02289527"},{"key":"e_1_2_1_7_2","doi-asserted-by":"publisher","DOI":"10.1002\/jgt.3190020304"},{"key":"e_1_2_1_8_2","doi-asserted-by":"publisher","DOI":"10.1002\/net.3230110302"},{"key":"e_1_2_1_9_2","doi-asserted-by":"publisher","DOI":"10.1287\/trsc.16.1.1"},{"key":"e_1_2_1_10_2","doi-asserted-by":"publisher","DOI":"10.1007\/BFb0073115"},{"key":"e_1_2_1_11_2","doi-asserted-by":"publisher","DOI":"10.1287\/mnsc.29.4.482"},{"key":"e_1_2_1_12_2","first-page":"87","article-title":"Medians and peripherians of trees","volume":"4","author":"Zelinka B.","year":"1968","journal-title":"Arch. Math. (Brno)"}],"container-title":["Networks"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/api.wiley.com\/onlinelibrary\/tdm\/v1\/articles\/10.1002%2Fnet.3230210103","content-type":"unspecified","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/onlinelibrary.wiley.com\/doi\/pdf\/10.1002\/net.3230210103","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2023,10,23]],"date-time":"2023-10-23T00:55:27Z","timestamp":1698022527000},"score":1,"resource":{"primary":{"URL":"https:\/\/onlinelibrary.wiley.com\/doi\/10.1002\/net.3230210103"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[1991,1]]},"references-count":11,"journal-issue":{"issue":"1","published-print":{"date-parts":[[1991,1]]}},"alternative-id":["10.1002\/net.3230210103"],"URL":"https:\/\/doi.org\/10.1002\/net.3230210103","archive":["Portico"],"relation":{},"ISSN":["0028-3045","1097-0037"],"issn-type":[{"value":"0028-3045","type":"print"},{"value":"1097-0037","type":"electronic"}],"subject":[],"published":{"date-parts":[[1991,1]]}}}