{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,12]],"date-time":"2025-10-12T02:25:29Z","timestamp":1760235929724,"version":"build-2065373602"},"reference-count":10,"publisher":"MDPI AG","issue":"10","license":[{"start":{"date-parts":[[2021,10,9]],"date-time":"2021-10-09T00:00:00Z","timestamp":1633737600000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"name":"Research Sector, Kuwait University","award":["SM05\/20"],"award-info":[{"award-number":["SM05\/20"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Symmetry"],"abstract":"<jats:p>Balanced signed graphs appear in the context of social groups with symmetric relations between individuals where a positive edge represents friendship and a negative edge represents enmities between the individuals. The frustration number f of a signed graph is the size of the minimal set F of vertices whose removal results in a balanced signed graph; hence, a connected signed graph G\u02d9 is balanced if and only if f=0. In this paper, we consider the balance of G\u02d9 via the relationships between the frustration number and eigenvalues of the symmetric Laplacian matrix associated with G\u02d9. It is known that a signed graph is balanced if and only if its least Laplacian eigenvalue \u03bcn is zero. We consider the inequalities that involve certain Laplacian eigenvalues, the frustration number f and some related invariants such as the cut size of F and its average vertex degree. In particular, we consider the interplay between \u03bcn and f.<\/jats:p>","DOI":"10.3390\/sym13101902","type":"journal-article","created":{"date-parts":[[2021,10,11]],"date-time":"2021-10-11T01:59:47Z","timestamp":1633917587000},"page":"1902","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["Inequalities for Laplacian Eigenvalues of Signed Graphs with Given Frustration Number"],"prefix":"10.3390","volume":"13","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-3348-1141","authenticated-orcid":false,"given":"Milica","family":"An\u0111eli\u0107","sequence":"first","affiliation":[{"name":"Department of Mathematics, Kuwait University, Safat 13060, Kuwait"}]},{"given":"Tamara","family":"Koledin","sequence":"additional","affiliation":[{"name":"School of Electrical Engineering, University of Belgrade, Bulevar Kralja Aleksandra 73, 11 000 Belgrade, Serbia"}]},{"given":"Zoran","family":"Stani\u0107","sequence":"additional","affiliation":[{"name":"Faculty of Mathematics, University of Belgrade, Studentski trg 16, 11 000 Belgrade, Serbia"}]}],"member":"1968","published-online":{"date-parts":[[2021,10,9]]},"reference":[{"key":"ref_1","unstructured":"Acharya, B.D., Katona, G.O.H., and Ne\u0161et\u0159il, J. (2010). Matrices in the theory of signed simple graphs. Advances in Discrete Mathematics and Applications: Mysore 2008, Ramanujan Mathematical Society."},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"47","DOI":"10.1016\/0166-218X(82)90033-6","article-title":"Signed graphs","volume":"4","author":"Zaslavsky","year":"1981","journal-title":"Discret. Appl. Math."},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"107","DOI":"10.1080\/00223980.1946.9917275","article-title":"Attitudes and cognitive organization","volume":"21","author":"Heider","year":"1946","journal-title":"J. Psychol."},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"133","DOI":"10.1016\/j.laa.2014.01.001","article-title":"Balancedness and the least eigenvalue of Laplacian of signed graphs","volume":"446","author":"Belardo","year":"2014","journal-title":"Linear Algebra Appl."},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"4165","DOI":"10.1007\/s00526-015-0935-x","article-title":"Frustration index and Cheeger inequalities for discrete and continuous magnetic Laplacians","volume":"54","author":"Lange","year":"2015","journal-title":"Calc. Var."},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"276","DOI":"10.1016\/j.dam.2016.09.015","article-title":"Frustration and isoperimetric inequalities for signed graphs","volume":"217","author":"Martin","year":"2017","journal-title":"Discret. Appl. Math."},{"key":"ref_7","doi-asserted-by":"crossref","unstructured":"Pasten, G., Rojo, O., and Medina, L. (2021). On the A\u03b1-eigenvalues of signed graphs. Mathematics, 9.","DOI":"10.3390\/math9161990"},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"80","DOI":"10.1016\/j.laa.2019.03.011","article-title":"Bounding the largest eigenvalue of signed graphs","volume":"573","year":"2019","journal-title":"Linear Algebra Appl."},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"119","DOI":"10.1016\/j.disc.2004.07.011","article-title":"Graphs and Hermitian matrices: Eigenvalue interlacing","volume":"289","author":"Nikiforov","year":"2004","journal-title":"Discret. Math."},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"227","DOI":"10.1016\/0166-218X(81)90001-9","article-title":"Balancing signed graphs","volume":"3","author":"Akiyama","year":"1981","journal-title":"Discret. Appl. Math."}],"container-title":["Symmetry"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2073-8994\/13\/10\/1902\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T07:10:59Z","timestamp":1760166659000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2073-8994\/13\/10\/1902"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2021,10,9]]},"references-count":10,"journal-issue":{"issue":"10","published-online":{"date-parts":[[2021,10]]}},"alternative-id":["sym13101902"],"URL":"https:\/\/doi.org\/10.3390\/sym13101902","relation":{},"ISSN":["2073-8994"],"issn-type":[{"type":"electronic","value":"2073-8994"}],"subject":[],"published":{"date-parts":[[2021,10,9]]}}}