{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T02:44:03Z","timestamp":1760150643449,"version":"build-2065373602"},"reference-count":30,"publisher":"MDPI AG","issue":"12","license":[{"start":{"date-parts":[[2023,12,15]],"date-time":"2023-12-15T00:00:00Z","timestamp":1702598400000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Axioms"],"abstract":"<jats:p>Hyperbolic triangle groups are found within the category of finitely generated groups. These are topological groups formed by the reflections along the sides of a hyperbolic triangle and acting properly discontinuously on the hyperbolic plane. Higman raised a question about the simplicity of finitely generated groups. The best known example of a simple group is the alternating group An, where n\u22655. This article establishes a relation between the hyperbolic triangle group denoted as \u25b5*(3,7,r) and the alternating group. The approach involves employing coset diagrams to establish this connection. The construction of adjacency matrices for these coset diagrams is performed, followed by a detailed examination of their spectral characteristics.<\/jats:p>","DOI":"10.3390\/axioms12121128","type":"journal-article","created":{"date-parts":[[2023,12,15]],"date-time":"2023-12-15T11:26:52Z","timestamp":1702639612000},"page":"1128","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["Spectral Analysis of the Adjacency Matrices for Alternating Quotients of Hyperbolic Triangle Group \u25b5*(3,q,r) for q &lt; r Primes"],"prefix":"10.3390","volume":"12","author":[{"given":"Sajida","family":"Younas","sequence":"first","affiliation":[{"name":"Department of Mathematics and Statistics, International Islamic University Islamabad, Islamabad 44000, Pakistan"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-8157-7909","authenticated-orcid":false,"given":"Sajida","family":"Kousar","sequence":"additional","affiliation":[{"name":"Department of Mathematics and Statistics, International Islamic University Islamabad, Islamabad 44000, Pakistan"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-6599-5975","authenticated-orcid":false,"given":"Majed","family":"Albaity","sequence":"additional","affiliation":[{"name":"Department of Mathematics, Faculty of Science, King Abdulaziz University, P.O. Box 80348, Jeddah 22254, Saudi Arabia"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-3871-3845","authenticated-orcid":false,"given":"Tahir","family":"Mahmood","sequence":"additional","affiliation":[{"name":"Department of Mathematics and Statistics, International Islamic University Islamabad, Islamabad 44000, Pakistan"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2023,12,15]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","unstructured":"Jones, G.A., and Singerman, D. (1987). Complex Functions an Algebraic and Geometric Viewpoint, Cambridge University Press.","DOI":"10.1017\/CBO9781139171915"},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"559","DOI":"10.1007\/s00208-020-02030-4","article-title":"Reflection maps","volume":"378","author":"Penafort","year":"2020","journal-title":"Math. 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