{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,11,4]],"date-time":"2025-11-04T09:47:56Z","timestamp":1762249676743},"reference-count":8,"publisher":"World Scientific Pub Co Pte Lt","issue":"04","content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Discrete Math. Algorithm. Appl."],"published-print":{"date-parts":[[2019,8]]},"abstract":"<jats:p> Let [Formula: see text] be a simple graph with [Formula: see text] vertices and [Formula: see text] be a sequence of [Formula: see text] rooted graphs [Formula: see text]. The rooted product [Formula: see text], of [Formula: see text] by [Formula: see text] is constructed by identifying the root vertex of [Formula: see text] with the [Formula: see text]th vertex of [Formula: see text]. In this paper, the characteristic polynomial of the normalized Laplacian matrix of [Formula: see text] is obtained. As an application of our results, we obtain the normalized Laplacian polynomial and spectrum of the generalized Bethe trees. <\/jats:p>","DOI":"10.1142\/s1793830919500460","type":"journal-article","created":{"date-parts":[[2019,7,17]],"date-time":"2019-07-17T04:16:35Z","timestamp":1563336995000},"page":"1950046","source":"Crossref","is-referenced-by-count":2,"title":["The normalized Laplacian polynomial of rooted product of graphs"],"prefix":"10.1142","volume":"11","author":[{"given":"Abbas","family":"Heydari","sequence":"first","affiliation":[{"name":"Department of Science, Arak University of Technology, Arak, Iran"}]}],"member":"219","published-online":{"date-parts":[[2019,9,16]]},"reference":[{"key":"S1793830919500460BIB001","doi-asserted-by":"publisher","DOI":"10.1080\/03081080902722741"},{"key":"S1793830919500460BIB002","doi-asserted-by":"publisher","DOI":"10.1016\/j.laa.2010.02.002"},{"key":"S1793830919500460BIB003","doi-asserted-by":"publisher","DOI":"10.1016\/j.dam.2006.09.008"},{"key":"S1793830919500460BIB004","volume-title":"CBMS Regional Conf. Series Mathematics","volume":"92","author":"Chung F. R. K.","year":"1997"},{"key":"S1793830919500460BIB005","doi-asserted-by":"publisher","DOI":"10.1017\/S0004972700007760"},{"key":"S1793830919500460BIB006","doi-asserted-by":"publisher","DOI":"10.1016\/j.laa.2008.05.016"},{"key":"S1793830919500460BIB007","doi-asserted-by":"publisher","DOI":"10.13001\/1081-3810.1195"},{"key":"S1793830919500460BIB008","doi-asserted-by":"publisher","DOI":"10.1016\/j.laa.2007.06.024"}],"container-title":["Discrete Mathematics, Algorithms and Applications"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.worldscientific.com\/doi\/pdf\/10.1142\/S1793830919500460","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2019,9,16]],"date-time":"2019-09-16T21:59:28Z","timestamp":1568671168000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.worldscientific.com\/doi\/abs\/10.1142\/S1793830919500460"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2019,8]]},"references-count":8,"journal-issue":{"issue":"04","published-print":{"date-parts":[[2019,8]]}},"alternative-id":["10.1142\/S1793830919500460"],"URL":"https:\/\/doi.org\/10.1142\/s1793830919500460","relation":{},"ISSN":["1793-8309","1793-8317"],"issn-type":[{"value":"1793-8309","type":"print"},{"value":"1793-8317","type":"electronic"}],"subject":[],"published":{"date-parts":[[2019,8]]}}}