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Unit dual quaternions represent rigid movements in the 3D space, and have wide applications in robotics and computer graphics. Dual complex numbers have found application in brain science recently. We establish the interlacing theorem for dual unit gain graphs, and show that the spectral radius of a dual unit gain graph is always not greater than the spectral radius of the underlying graph, and these two radii are equal if, and only if, the dual gain graph is balanced. By using dual cosine functions, we establish the closed form of the eigenvalues of adjacency and Laplacian matrices of dual complex and quaternion unit gain cycles. We then show the coefficient theorem holds for dual unit gain graphs. Similar results hold for the spectral radius of the Laplacian matrix of the dual unit gain graph too.<\/jats:p>","DOI":"10.3390\/sym16091142","type":"journal-article","created":{"date-parts":[[2024,9,3]],"date-time":"2024-09-03T08:38:47Z","timestamp":1725352727000},"page":"1142","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":9,"title":["Spectral Properties of Dual Unit Gain Graphs"],"prefix":"10.3390","volume":"16","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-6815-4060","authenticated-orcid":false,"given":"Chunfeng","family":"Cui","sequence":"first","affiliation":[{"name":"LMIB of the Ministry of Education, School of Mathematical Sciences, Beihang University, Beijing 100191, China"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Yong","family":"Lu","sequence":"additional","affiliation":[{"name":"School of Mathematics and Statistics, Jiangsu Normal University, Xuzhou 221116, China"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-1112-5250","authenticated-orcid":false,"given":"Liqun","family":"Qi","sequence":"additional","affiliation":[{"name":"Department of Applied Mathematics, The Hong Kong Polytechnic University, Hung Hom, Kowloon, Hong Kong"},{"name":"Department of Mathematics, School of Science, Hangzhou Dianzi University, Hangzhou 310018, China"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Ligong","family":"Wang","sequence":"additional","affiliation":[{"name":"School of Mathematics and Statistics, Northwestern Polytechnical University, Xi\u2019an 710072, China"}],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"1968","published-online":{"date-parts":[[2024,9,3]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"265","DOI":"10.1007\/s10801-020-00977-w","article-title":"A group representation approach to balance of gain graphs","volume":"54","author":"Cavaleri","year":"2021","journal-title":"J. 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