{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,7,30]],"date-time":"2025-07-30T14:24:07Z","timestamp":1753885447087,"version":"3.41.2"},"reference-count":17,"publisher":"World Scientific Pub Co Pte Ltd","issue":"05","content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Discrete Math. Algorithm. Appl."],"published-print":{"date-parts":[[2023,7]]},"abstract":"<jats:p> For a [Formula: see text]-regular graph [Formula: see text], an edge-signing [Formula: see text] is called a good signing if the absolute eigenvalues of its adjacency matrix are at most [Formula: see text]. Bilu and Linial conjectured that for each regular graph there exists a good signing. In this paper, by using the concept of \u201cEquitable Partition\u201d, we prove the conjecture for some cases. We show how to find out a good signing for special complete graphs and lexicographic product of two graphs. In particular, if there exist two good signings for graph [Formula: see text], then we can find a good signing for a 2-lift of [Formula: see text]. <\/jats:p>","DOI":"10.1142\/s1793830922501294","type":"journal-article","created":{"date-parts":[[2022,7,26]],"date-time":"2022-07-26T08:44:39Z","timestamp":1658825079000},"source":"Crossref","is-referenced-by-count":0,"title":["Equitable partition for some Ramanujan graphs"],"prefix":"10.1142","volume":"15","author":[{"ORCID":"https:\/\/orcid.org\/0000-0001-6818-3753","authenticated-orcid":false,"given":"M.","family":"Alinejad","sequence":"first","affiliation":[{"name":"Department of Pure Mathematics, Ferdowsi University of Mashhad, P.O. Box 1159-91775, Mashhad, Iran"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"A.","family":"Erfanian","sequence":"additional","affiliation":[{"name":"Department of Pure Mathematics, Ferdowsi University of Mashhad, P.O. Box 1159-91775, Mashhad, Iran"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"S.","family":"Fulad","sequence":"additional","affiliation":[{"name":"Department of Mathematics, School of Science, Shiraz University, P.O. Box 7146713565, Shiraz, Iran"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"219","published-online":{"date-parts":[[2022,8,11]]},"reference":[{"issue":"1","key":"S1793830922501294BIB001","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1002\/rsa.10003","volume":"20","author":"Amit A.","year":"2002","journal-title":"Random Struct. Algorithms"},{"issue":"5","key":"S1793830922501294BIB002","doi-asserted-by":"crossref","first-page":"495","DOI":"10.1007\/s00493-006-0029-7","volume":"26","author":"Bilu Y.","year":"2006","journal-title":"Combinatorica"},{"key":"S1793830922501294BIB003","first-page":"1","volume":"57","author":"Burnwal S. P.","year":"2021","journal-title":"Ramanujan J."},{"key":"S1793830922501294BIB004","doi-asserted-by":"crossref","first-page":"275","DOI":"10.1007\/BF01285816","volume":"12","author":"Chiu P.","year":"1992","journal-title":"Combinatorica"},{"issue":"1","key":"S1793830922501294BIB005","doi-asserted-by":"crossref","first-page":"19","DOI":"10.1215\/S0012-7094-03-11812-8","volume":"118","author":"Friedman J.","year":"2003","journal-title":"Duke Math. J."},{"key":"S1793830922501294BIB006","doi-asserted-by":"crossref","DOI":"10.1007\/978-1-4613-0163-9","volume-title":"Algebraic Graph Theory","author":"Godsil C.","year":"2001"},{"key":"S1793830922501294BIB008","doi-asserted-by":"crossref","DOI":"10.1017\/CBO9780511810817","volume-title":"Matrix Analysis","author":"Horn R. A.","year":"1985"},{"issue":"2","key":"S1793830922501294BIB009","doi-asserted-by":"crossref","first-page":"453","DOI":"10.1007\/s10623-019-00692-z","volume":"88","author":"Hyun J. Y.","year":"2020","journal-title":"Des. Codes Cryptogr."},{"issue":"5","key":"S1793830922501294BIB010","doi-asserted-by":"crossref","first-page":"1007","DOI":"10.1016\/S0040-9383(96)00044-4","volume":"36","author":"Jordan B. W.","year":"1997","journal-title":"Topology"},{"key":"S1793830922501294BIB011","series-title":"Society for Industrial and Applied Mathematics","first-page":"2230","volume-title":"Proc. 28th Annual ACM-SIAM Symp. Discrete Algorithms","author":"Lelarge M.","year":"2017"},{"issue":"4","key":"S1793830922501294BIB012","doi-asserted-by":"crossref","first-page":"1612","DOI":"10.1016\/j.aim.2011.03.016","volume":"227","author":"Lubetzky E.","year":"2011","journal-title":"Adv. Math."},{"key":"S1793830922501294BIB013","doi-asserted-by":"crossref","first-page":"261","DOI":"10.1007\/BF02126799","volume":"8","author":"Lubotzky A.","year":"1988","journal-title":"Combinatorica"},{"key":"S1793830922501294BIB014","first-page":"51","volume":"24","author":"Margulis G. A.","year":"1988","journal-title":"Probl. Peredachi Inf."},{"key":"S1793830922501294BIB015","doi-asserted-by":"crossref","first-page":"529","DOI":"10.1109\/FOCS.2013.63","volume-title":"2013 IEEE 54th Ann. Symp. Foundations of Computer Science","author":"Marcus A.","year":"2013"},{"key":"S1793830922501294BIB016","doi-asserted-by":"crossref","first-page":"44","DOI":"10.1006\/jctb.1994.1054","volume":"62","author":"Morgenstern M.","year":"1994","journal-title":"J. Combin. Theory Ser. B"},{"key":"S1793830922501294BIB017","doi-asserted-by":"crossref","first-page":"127","DOI":"10.1090\/S0273-0979-1990-15918-X","volume":"23","author":"Pizer A. K.","year":"1990","journal-title":"Bull. Amer. Math. Soc."},{"volume-title":"A Course in Combinatorics","year":"1992","author":"Van Lint J.","key":"S1793830922501294BIB018"}],"container-title":["Discrete Mathematics, Algorithms and Applications"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.worldscientific.com\/doi\/pdf\/10.1142\/S1793830922501294","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2023,5,23]],"date-time":"2023-05-23T03:17:40Z","timestamp":1684811860000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.worldscientific.com\/doi\/10.1142\/S1793830922501294"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2022,8,11]]},"references-count":17,"journal-issue":{"issue":"05","published-print":{"date-parts":[[2023,7]]}},"alternative-id":["10.1142\/S1793830922501294"],"URL":"https:\/\/doi.org\/10.1142\/s1793830922501294","relation":{},"ISSN":["1793-8309","1793-8317"],"issn-type":[{"type":"print","value":"1793-8309"},{"type":"electronic","value":"1793-8317"}],"subject":[],"published":{"date-parts":[[2022,8,11]]},"article-number":"2250129"}}