{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,12]],"date-time":"2025-10-12T00:28:27Z","timestamp":1760228907495,"version":"build-2065373602"},"reference-count":28,"publisher":"MDPI AG","issue":"6","license":[{"start":{"date-parts":[[2022,5,27]],"date-time":"2022-05-27T00:00:00Z","timestamp":1653609600000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"name":"College of Arts and Sciences, Stetson University"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Symmetry"],"abstract":"<jats:p>A square array whose all rows and columns are different permutations of the same length over the same symbol set is known as a Latin square. A Latin square may or may not be symmetric. For classification and enumeration purposes, symmetric, non-symmetric, conjugate symmetric, and totally symmetric Latin squares play vital roles. This article discusses the Eigenproblem of non-symmetric Latin squares in well known max-plus algebra. By defining a certain vector corresponding to each cycle of a permutation of the Latin square, we characterize and find the Eigenvalue as well as the possible Eigenvectors.<\/jats:p>","DOI":"10.3390\/sym14061101","type":"journal-article","created":{"date-parts":[[2022,5,31]],"date-time":"2022-05-31T02:30:06Z","timestamp":1653964206000},"page":"1101","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["Trivial and Nontrivial Eigenvectors for Latin Squares in Max-Plus Algebra"],"prefix":"10.3390","volume":"14","author":[{"given":"Fazal","family":"Abbas","sequence":"first","affiliation":[{"name":"Department of Mathematics and Computer Sciences, Stetson University, DeLand, FL 32723, USA"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-6523-4316","authenticated-orcid":false,"given":"Mubasher","family":"Umer","sequence":"additional","affiliation":[{"name":"Department of Mathematics, Quaid-i-Azam University, Islamabad 45320, Pakistan"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-1373-7701","authenticated-orcid":false,"given":"Umar","family":"Hayat","sequence":"additional","affiliation":[{"name":"Department of Mathematics, Quaid-i-Azam University, Islamabad 45320, Pakistan"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-4431-3186","authenticated-orcid":false,"given":"Ikram","family":"Ullah","sequence":"additional","affiliation":[{"name":"Department of Mathematics, Quaid-i-Azam University, Islamabad 45320, Pakistan"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2022,5,27]]},"reference":[{"key":"ref_1","first-page":"887","article-title":"Tropical nevanlinna theory and ultra-discrete equations","volume":"5","author":"Halburd","year":"2009","journal-title":"Int. Math. Res. Not."},{"key":"ref_2","doi-asserted-by":"crossref","unstructured":"Cuninghame-Green, R.A. (1979). Lecture notes in economics and mathematical systems. Minimax Algebra, Springer.","DOI":"10.1007\/978-3-642-48708-8"},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"103","DOI":"10.1016\/S0024-3795(00)00013-6","article-title":"On the ultimate behavior of the sequence of consecutive powers of a matrix in the max-plus algebra","volume":"307","year":"2000","journal-title":"Linear Algebra Its Appl."},{"key":"ref_4","doi-asserted-by":"crossref","unstructured":"Gaubert, S. (1997). Methods and applications of (max,+) linear algebra. Annual Symposium on Theoretical Aspects of Computer Science, Springer.","DOI":"10.1007\/BFb0023465"},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"6718653","DOI":"10.1155\/2018\/6718653","article-title":"On max-plus algebra and its application on image steganography","volume":"2018","author":"Santoso","year":"2018","journal-title":"Sci. World J."},{"key":"ref_6","first-page":"627","article-title":"A cryptographic algorithm using wavelet transforms over max-plus algebra","volume":"34","author":"Cahyono","year":"2022","journal-title":"J. King Saud-Univ.-Comput. Inf. Sci."},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"87","DOI":"10.1016\/0024-3795(88)90145-0","article-title":"Cram\u00e9r and Cayley-Hamilton in the max algebra","volume":"101","author":"Olsder","year":"1988","journal-title":"Linear Algebra Its Appl."},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"237","DOI":"10.1016\/S0024-3795(96)00407-7","article-title":"A note on the characteristic equation in the max-plus algebra","volume":"261","year":"1997","journal-title":"Linear Algebra Its Appl."},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"109104","DOI":"10.1016\/j.automatica.2020.109104","article-title":"Global optimization for max-plus linear systems and applications in distributed systems","volume":"119","author":"Tao","year":"2020","journal-title":"Automatica"},{"key":"ref_10","doi-asserted-by":"crossref","unstructured":"Marotta, A.M., Gon\u00e7alves, V.M., and Maia, C.A. (2020). Tropical lexicographic optimization: Synchronizing timed event graphs. Symmetry, 12.","DOI":"10.3390\/sym12101597"},{"key":"ref_11","doi-asserted-by":"crossref","unstructured":"My\u0161kov\u00e1, H., and Plavka, J. (2021). Polynomial and pseudopolynomial procedures for solving interval two-sided (max, plus)-linear systems. Mathematics, 9.","DOI":"10.3390\/math9222951"},{"key":"ref_12","doi-asserted-by":"crossref","unstructured":"Wang, C., Xia, Y., and Tao, Y. (2021). Ordered structures of polynomials over max-plus algebra. Symmetry, 13.","DOI":"10.3390\/sym13071137"},{"key":"ref_13","unstructured":"(2000). On Classes of Min-Max-Plus Systems and Their Application. [Ph.D. Thesis, Delft University of Technology]."},{"key":"ref_14","first-page":"1932361","article-title":"A max-plus algebra approach to study time disturbance propagation within a robustness improvement context","volume":"2018","year":"2018","journal-title":"Math. Probl. Eng."},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"189","DOI":"10.1016\/S0304-3975(02)00237-2","article-title":"Application of max-plus algebra to biological sequence comparisons","volume":"293","author":"Comet","year":"2003","journal-title":"Theor. Comput. Sci."},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"596","DOI":"10.1016\/j.dam.2007.07.015","article-title":"Structure of the eigenspace of a Monge matrix in max-plus algebra","volume":"10","author":"Gavalec","year":"2008","journal-title":"Discret. Appl. Math."},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"1701","DOI":"10.1016\/j.dam.2010.06.008","article-title":"Computing an eigenvector of an inverse Monge matrix in maxplus algebra","volume":"158","author":"Imaev","year":"2010","journal-title":"Discret. Appl. Math."},{"key":"ref_18","unstructured":"Tomaskova, H. (2010, January 3\u20136). Eigenproblem for circulant matrices in max-plus algebra. Proceedings of the 12th WSEAS international conference on Mathematical Methods, Computational Techniques and Intelligent Systems, Sousse, Tunisia."},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"369","DOI":"10.1023\/A:1008315821604","article-title":"Power algorithms for (max,+)- and bipartite (min,max,+)-systems","volume":"10","author":"Subiono","year":"2000","journal-title":"Discret. Event Dyn. Syst."},{"key":"ref_20","doi-asserted-by":"crossref","unstructured":"Umer, M., Hayat, U., and Abbas, F. (2019). An efficient algorithm for nontrivial eigenvectors in max-plus algebra. Symmetry, 11.","DOI":"10.3390\/sym11060738"},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"37","DOI":"10.22342\/jims.20.1.178.37-45","article-title":"Subiono Eigenvalues and eigenvectors of latin squares in max-plus algebra","volume":"20","author":"Mufid","year":"2014","journal-title":"J. Indones. Math. Soc."},{"key":"ref_22","doi-asserted-by":"crossref","unstructured":"Umer, M., Hayat, U., Abbas, F., Agarwal, A., and Kitanov, P. (2020). An efficient algorithm for eigenvalue problem of Latin squares in a bipartite min-max-plus system. Symmetry, 12.","DOI":"10.3390\/sym12020311"},{"key":"ref_23","doi-asserted-by":"crossref","first-page":"3261","DOI":"10.1016\/j.laa.2011.06.009","article-title":"Best approximation in maxplus semimodules","volume":"435","author":"Akian","year":"2011","journal-title":"Linear Algebra Its Appl."},{"key":"ref_24","doi-asserted-by":"crossref","first-page":"67","DOI":"10.1016\/0024-3795(93)90492-7","article-title":"The power algorithm in max algebra","volume":"182","author":"Braker","year":"1993","journal-title":"Linear Algebra Its Appl."},{"key":"ref_25","doi-asserted-by":"crossref","first-page":"390","DOI":"10.4236\/apm.2015.57038","article-title":"Eigenvectors of permutation matrices","volume":"5","author":"Magret","year":"2015","journal-title":"Adv. Pure Math."},{"key":"ref_26","unstructured":"Hanniah, U. (2022, April 25). Subvektor Eigen Bilangan Bulat Dalam Aljabar Maks-Plus. Available online: https:\/\/digilib.uns.ac.id\/dokumen\/download\/80035\/NDMxNTQ2\/Subvektor-Eigen-Bilangan-Bulat-Dalam-Aljabar-Maks-Plus-abstrak.pdf."},{"key":"ref_27","doi-asserted-by":"crossref","unstructured":"Rosyada, S.A., and Kurniawan, S.V.Y. (2021, January 16\u201318). Bases in min-plus algebra. Proceedings of the International Conference of Mathematics and Mathematics Education (I-CMME 2021), Ankara, Turkey.","DOI":"10.2991\/assehr.k.211122.044"},{"key":"ref_28","doi-asserted-by":"crossref","first-page":"334","DOI":"10.1007\/s00026-005-0261-7","article-title":"On the number of Latin squares","volume":"9","author":"McKay","year":"2005","journal-title":"Ann. Comb."}],"container-title":["Symmetry"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2073-8994\/14\/6\/1101\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,10]],"date-time":"2025-10-10T23:19:48Z","timestamp":1760138388000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2073-8994\/14\/6\/1101"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2022,5,27]]},"references-count":28,"journal-issue":{"issue":"6","published-online":{"date-parts":[[2022,6]]}},"alternative-id":["sym14061101"],"URL":"https:\/\/doi.org\/10.3390\/sym14061101","relation":{},"ISSN":["2073-8994"],"issn-type":[{"type":"electronic","value":"2073-8994"}],"subject":[],"published":{"date-parts":[[2022,5,27]]}}}