{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,11,7]],"date-time":"2025-11-07T11:58:04Z","timestamp":1762516684991,"version":"build-2065373602"},"reference-count":15,"publisher":"MDPI AG","issue":"11","license":[{"start":{"date-parts":[[2025,11,7]],"date-time":"2025-11-07T00:00:00Z","timestamp":1762473600000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"DOI":"10.13039\/100009110","name":"Natural Science Foundation of Xinjiang Uygur Autonomous Region","doi-asserted-by":"publisher","award":["2024D01C41"],"award-info":[{"award-number":["2024D01C41"]}],"id":[{"id":"10.13039\/100009110","id-type":"DOI","asserted-by":"publisher"}]},{"name":"Tianshan Talent Training Program","award":["2024TSYCCX0013"],"award-info":[{"award-number":["2024TSYCCX0013"]}]},{"DOI":"10.13039\/501100001809","name":"NSFC","doi-asserted-by":"publisher","award":["12361071","11901498"],"award-info":[{"award-number":["12361071","11901498"]}],"id":[{"id":"10.13039\/501100001809","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Axioms"],"abstract":"<jats:p>For a connected graph G, let DL(G) be its distance Laplacian matrix and \u03bb1(G)\u2265\u03bb2(G)\u2265\u2026\u2265\u03bbn\u22121(G)&gt;\u03bbn(G)=0 be its DL eigenvalues. The DL energy of G is defined as DLEG=\u2211i=1n\u03bbi(G)\u22122WGn, where W(G) is the Wiener index of G. An important problem in graph energy studies is to determine exact formulations of the energy for specific graph classes and their complements. This paper gives the precise DL energy formulations of a class of bicyclic graphs C2(p,q), a class of unicyclic graphs C1(p,q), and their complements. Moreover, we order the graphs C2(p,q) on the basic of \u03bb1, \u03bbn\u22121, and consider the same problems for their complements. And the ordering of the graphs C1(p,q) on the basic of \u03bbn\u22121 and the ordering of their complements on the basics of \u03bb1, \u03bbn\u22121 and the DL energy are obtained.<\/jats:p>","DOI":"10.3390\/axioms14110825","type":"journal-article","created":{"date-parts":[[2025,11,7]],"date-time":"2025-11-07T11:41:47Z","timestamp":1762515707000},"page":"825","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["On Distance Laplacian Energy of Unicyclic and Bicyclic Graphs"],"prefix":"10.3390","volume":"14","author":[{"given":"Dan","family":"Li","sequence":"first","affiliation":[{"name":"College of Mathematics and System Science, Xinjiang University, Urumqi 830046, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Shiqi","family":"Zhou","sequence":"additional","affiliation":[{"name":"College of Mathematics and System Science, Xinjiang University, Urumqi 830046, China"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2025,11,7]]},"reference":[{"key":"ref_1","first-page":"1","article-title":"The energy of a graph","volume":"103","author":"Gutman","year":"1978","journal-title":"Ber. Math. Statist. Sekt. Forschungsz. Graz."},{"key":"ref_2","first-page":"3","article-title":"On energy of graphs","volume":"77","author":"Das","year":"2017","journal-title":"MATCH Commun. Math. Comput. Chem."},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"79","DOI":"10.1016\/j.laa.2016.03.003","article-title":"On energy of line graphs","volume":"499","author":"Das","year":"2016","journal-title":"Linear Algebra Its Appl."},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"133","DOI":"10.1016\/j.laa.2016.07.007","article-title":"Remarks on the energy of regular graphs","volume":"508","author":"Nikiforov","year":"2016","journal-title":"Linear Algebra Its Appl."},{"key":"ref_5","first-page":"261","article-title":"Energy of monad graphs","volume":"11","author":"Shukur","year":"2021","journal-title":"Bull. Int. Math. Virtual Inst."},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"172","DOI":"10.1016\/j.dam.2018.01.004","article-title":"On (distance) Laplacian energy and (distance) signless Laplacian energy of graphs","volume":"243","author":"Das","year":"2018","journal-title":"Discret. Appl. Math."},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"55","DOI":"10.1016\/j.laa.2018.01.032","article-title":"Sharp upper bounds on the distance energies of a graph","volume":"545","author":"Diaz","year":"2018","journal-title":"Linear Algebra Its Appl."},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"918","DOI":"10.1016\/j.aml.2010.12.050","article-title":"Laplacian energy of diameter 3 trees","volume":"24","author":"Trevisan","year":"2011","journal-title":"Appl. Math. Lett."},{"key":"ref_9","first-page":"245","article-title":"Bounds on the DL energy of graphs","volume":"37","author":"Yang","year":"2013","journal-title":"Kragujev. J. Math."},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"867","DOI":"10.1007\/s12190-024-01995-8","article-title":"On distance Laplacian spectral ordering of some graphs","volume":"70","author":"Rather","year":"2024","journal-title":"J. Appl. Math. Comput."},{"key":"ref_11","first-page":"127847","article-title":"Distance Laplacian spectral ordering of sun type graphs","volume":"445","author":"Rather","year":"2023","journal-title":"Appl. Math. Comput."},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"2050061","DOI":"10.1142\/S1793830920500615","article-title":"On DL spectrum (energy) of graphs","volume":"12","author":"Ganie","year":"2020","journal-title":"Discret. Math. Algorithms Appl."},{"key":"ref_13","first-page":"125762","article-title":"On the DL energy ordering of trees","volume":"394","author":"Ganie","year":"2021","journal-title":"Appl. Math. Comput."},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"751","DOI":"10.1007\/s10587-014-0129-2","article-title":"Some properties of DL eigenvalues of a graph","volume":"64","author":"Aouchiche","year":"2014","journal-title":"Czechoslov. Math. J."},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"97","DOI":"10.1016\/j.laa.2014.07.025","article-title":"On DL spectra of graphs","volume":"460","author":"Nath","year":"2014","journal-title":"Linear Algebra Its Appl."}],"container-title":["Axioms"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2075-1680\/14\/11\/825\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,11,7]],"date-time":"2025-11-07T11:53:12Z","timestamp":1762516392000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2075-1680\/14\/11\/825"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2025,11,7]]},"references-count":15,"journal-issue":{"issue":"11","published-online":{"date-parts":[[2025,11]]}},"alternative-id":["axioms14110825"],"URL":"https:\/\/doi.org\/10.3390\/axioms14110825","relation":{},"ISSN":["2075-1680"],"issn-type":[{"type":"electronic","value":"2075-1680"}],"subject":[],"published":{"date-parts":[[2025,11,7]]}}}