{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,2,24]],"date-time":"2026-02-24T07:03:57Z","timestamp":1771916637518,"version":"3.50.1"},"reference-count":12,"publisher":"MDPI AG","issue":"10","license":[{"start":{"date-parts":[[2019,10,5]],"date-time":"2019-10-05T00:00:00Z","timestamp":1570233600000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"DOI":"10.13039\/501100006193","name":"College of Graduate Studies, Kuwait University","doi-asserted-by":"publisher","award":["SM03\/18"],"award-info":[{"award-number":["SM03\/18"]}],"id":[{"id":"10.13039\/501100006193","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Mathematics"],"abstract":"<jats:p>Let      T 4  =  { \u00b1 1 , \u00b1 i }      be the subgroup of fourth roots of unity inside    T   , the multiplicative group of complex units. For a     T 4    -gain graph     \u03a6 = ( \u0393 ,  T 4  , \u03c6 )    , we introduce gain functions on its line graph     L ( \u0393 )     and on its subdivision graph     S ( \u0393 )    . The corresponding gain graphs     L ( \u03a6 )     and     S ( \u03a6 )     are defined up to switching equivalence and generalize the analogous constructions for signed graphs. We discuss some spectral properties of these graphs and in particular we establish the relationship between the Laplacian characteristic polynomial of a gain graph    \u03a6   , and the adjacency characteristic polynomials of     L ( \u03a6 )     and     S ( \u03a6 )    . A suitably defined incidence matrix for     T 4    -gain graphs plays an important role in this context.<\/jats:p>","DOI":"10.3390\/math7100926","type":"journal-article","created":{"date-parts":[[2019,10,7]],"date-time":"2019-10-07T03:34:01Z","timestamp":1570419241000},"page":"926","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":14,"title":["Line and Subdivision Graphs Determined by \r\n          \r\n            \r\n              \r\n                \r\n                  T\r\n                  4\r\n                \r\n              \r\n            \r\n          \r\n        -Gain Graphs"],"prefix":"10.3390","volume":"7","author":[{"ORCID":"https:\/\/orcid.org\/0000-0001-9383-3121","authenticated-orcid":false,"given":"Abdullah","family":"Alazemi","sequence":"first","affiliation":[{"name":"Department of Mathematics, Kuwait University, Safat 13060, Kuwait"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-3348-1141","authenticated-orcid":false,"given":"Milica","family":"An\u0111eli\u0107","sequence":"additional","affiliation":[{"name":"Department of Mathematics, Kuwait University, Safat 13060, Kuwait"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-4253-2905","authenticated-orcid":false,"given":"Francesco","family":"Belardo","sequence":"additional","affiliation":[{"name":"Department of Mathematics and Applications, University of Naples, 80138 Napoli, Italy"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-2742-1919","authenticated-orcid":false,"given":"Maurizio","family":"Brunetti","sequence":"additional","affiliation":[{"name":"Department of Mathematics and Applications, University of Naples, 80138 Napoli, Italy"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-7742-4416","authenticated-orcid":false,"given":"Carlos M.","family":"da Fonseca","sequence":"additional","affiliation":[{"name":"Kuwait College of Science and Technology, Doha District, Block 4, P.O. Box 27235, Safat 13133, Kuwait"},{"name":"University of Primorska, FAMNIT, Glagoljsa\u0161ka 8, 6000 Koper, Slovenia"}]}],"member":"1968","published-online":{"date-parts":[[2019,10,5]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"32","DOI":"10.1016\/0095-8956(89)90063-4","article-title":"Biased graphs. I: Bias, balance, and gains","volume":"47","author":"Zaslavsky","year":"1989","journal-title":"J. Combin. Theory Ser. B"},{"key":"ref_2","unstructured":"Zaslavsky, T. (1998). A mathematical bibliography of signed and gain graphs and allied areas. Electron. J. Comb., 5, Available online: https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/view\/DS8."},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"3165","DOI":"10.1016\/j.laa.2011.10.021","article-title":"Spectral properties of complex unit gain graphs","volume":"436","author":"Reff","year":"2012","journal-title":"Linear Algebra Appl."},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"316","DOI":"10.1016\/j.laa.2016.05.040","article-title":"Oriented gain graphs, line graphs and eigenvalues","volume":"506","author":"Reff","year":"2016","journal-title":"Linear Algebra Appl."},{"key":"ref_5","doi-asserted-by":"crossref","unstructured":"Belardo, F., Brunetti, M., and Reff, N. (2020). Balancedness and the least Laplacian eigenvalue of some complex unit gain graphs. Discuss. Math. Graph Theory, in press.","DOI":"10.7151\/dmgt.2281"},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1016\/j.laa.2016.08.011","article-title":"Laplacian matrices of general complex weighted directed graphs","volume":"510","author":"Dong","year":"2016","journal-title":"Linear Algebra Appl."},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"227","DOI":"10.13001\/1081-3810.3029","article-title":"Properties of first eigenvectors and first eigenvalues of nonsingular weighted directed graphs","volume":"30","author":"Kalita","year":"2015","journal-title":"Electron. J. 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Graph Theory"},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"94","DOI":"10.1016\/j.laa.2015.02.007","article-title":"On the Laplacian coefficients of signed graphs","volume":"475","author":"Belardo","year":"2015","journal-title":"Linear Algebra Appl."},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"144","DOI":"10.1016\/j.laa.2015.04.022","article-title":"Spectral characterizations of signed lollipop graphs","volume":"480","author":"Belardo","year":"2015","journal-title":"Linear Algebra Appl."},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"81","DOI":"10.1016\/j.disc.2017.07.003","article-title":"On the determinant of the Laplacian matrix of a complex unit gain graph","volume":"341","author":"Wang","year":"2018","journal-title":"Discrete Math."}],"container-title":["Mathematics"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2227-7390\/7\/10\/926\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T13:27:48Z","timestamp":1760189268000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2227-7390\/7\/10\/926"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2019,10,5]]},"references-count":12,"journal-issue":{"issue":"10","published-online":{"date-parts":[[2019,10]]}},"alternative-id":["math7100926"],"URL":"https:\/\/doi.org\/10.3390\/math7100926","relation":{},"ISSN":["2227-7390"],"issn-type":[{"value":"2227-7390","type":"electronic"}],"subject":[],"published":{"date-parts":[[2019,10,5]]}}}