{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,12]],"date-time":"2025-10-12T02:49:24Z","timestamp":1760237364912,"version":"build-2065373602"},"reference-count":12,"publisher":"MDPI AG","issue":"5","license":[{"start":{"date-parts":[[2020,5,13]],"date-time":"2020-05-13T00:00:00Z","timestamp":1589328000000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Symmetry"],"abstract":"<jats:p>The line graph of a graph G is another graph of which the vertex set corresponds to the edge set of G, and two vertices of the line graph of G are adjacent if the corresponding edges in G share a common vertex. A graph is reflexive if the second-largest eigenvalue of its adjacency matrix is no greater than 2. Reflexive graphs give combinatorial ground to generate two classes of algebraic numbers, Salem and Pisot numbers. The difficult question of identifying those graphs whose line graphs are reflexive (called L-reflexive graphs) is naturally attacked by first answering this question for trees. Even then, however, an elegant full characterization of reflexive line graphs of trees has proved to be quite formidable. In this paper, we present an efficient algorithm for the exhaustive generation of maximal L-reflexive trees.<\/jats:p>","DOI":"10.3390\/sym12050809","type":"journal-article","created":{"date-parts":[[2020,5,14]],"date-time":"2020-05-14T02:55:41Z","timestamp":1589424941000},"page":"809","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":1,"title":["Efficient Algorithm for Generating Maximal L-Reflexive Trees"],"prefix":"10.3390","volume":"12","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"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-2400-6020","authenticated-orcid":false,"given":"Dejan","family":"\u017divkovi\u0107","sequence":"additional","affiliation":[{"name":"Faculty of Informatics and Computing, Singidunum University, 11000 Belgrade, Serbia"}]}],"member":"1968","published-online":{"date-parts":[[2020,5,13]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","unstructured":"Cvetkovi\u0107, D., Rowlinson, P., and Simi\u0107, S. (2004). Spectral Generalizations of Line Graphs: On Graphs with Least Eigenvalue \u22122, Cambridge University Press.","DOI":"10.1017\/CBO9780511751752"},{"key":"ref_2","doi-asserted-by":"crossref","unstructured":"Cvetkovi\u0107, D., Rowlinson, P., and Simi\u0107, S. (2009). An Introduction to the Theory of Graph Spectra, Cambridge University Press.","DOI":"10.1017\/CBO9780511801518"},{"key":"ref_3","first-page":"504","article-title":"Graphs with least eigenvalue \u22122: Ten years on","volume":"483","author":"Rowlinson","year":"2015","journal-title":"Linear Algebra Appl."},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"447","DOI":"10.1007\/s10801-015-0640-z","article-title":"Reflexive line graphs of trees","volume":"43","year":"2016","journal-title":"J. Algebraic Combin."},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"262","DOI":"10.1016\/j.laa.2014.09.032","article-title":"Notes on the second largest eigenvalue of a graph","volume":"465","year":"2015","journal-title":"Linear Algebra Appl."},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"46","DOI":"10.1016\/0021-8693(78)90021-2","article-title":"Hyperbolic trees","volume":"54","author":"Maxwell","year":"1978","journal-title":"J. Algebra"},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"2","DOI":"10.1016\/0024-3795(82)90022-2","article-title":"The second largest eigenvalue of trees","volume":"46","author":"Neumaier","year":"1982","journal-title":"Linear Algebra Appl."},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"228","DOI":"10.2298\/AADM0701228R","article-title":"On unicyclic reflexive graphs","volume":"1","year":"2007","journal-title":"Appl. Anal. 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Math."}],"container-title":["Symmetry"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2073-8994\/12\/5\/809\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T09:28:16Z","timestamp":1760174896000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2073-8994\/12\/5\/809"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2020,5,13]]},"references-count":12,"journal-issue":{"issue":"5","published-online":{"date-parts":[[2020,5]]}},"alternative-id":["sym12050809"],"URL":"https:\/\/doi.org\/10.3390\/sym12050809","relation":{},"ISSN":["2073-8994"],"issn-type":[{"type":"electronic","value":"2073-8994"}],"subject":[],"published":{"date-parts":[[2020,5,13]]}}}