{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T01:49:48Z","timestamp":1760147388946,"version":"build-2065373602"},"reference-count":23,"publisher":"MDPI AG","issue":"3","license":[{"start":{"date-parts":[[2023,2,1]],"date-time":"2023-02-01T00:00:00Z","timestamp":1675209600000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"name":"Centre for Mathematics from the University of Coimbra","award":["UIDB\/00324\/2020","FCT\/MCTES","Teor\u00eda de Lie y Teor\u00eda de Espacios de Banach","FQM298","FEDER-UCA18-107643"],"award-info":[{"award-number":["UIDB\/00324\/2020","FCT\/MCTES","Teor\u00eda de Lie y Teor\u00eda de Espacios de Banach","FQM298","FEDER-UCA18-107643"]}]},{"name":"Portuguese Government","award":["UIDB\/00324\/2020","FCT\/MCTES","Teor\u00eda de Lie y Teor\u00eda de Espacios de Banach","FQM298","FEDER-UCA18-107643"],"award-info":[{"award-number":["UIDB\/00324\/2020","FCT\/MCTES","Teor\u00eda de Lie y Teor\u00eda de Espacios de Banach","FQM298","FEDER-UCA18-107643"]}]},{"name":"PCI of the UCA","award":["UIDB\/00324\/2020","FCT\/MCTES","Teor\u00eda de Lie y Teor\u00eda de Espacios de Banach","FQM298","FEDER-UCA18-107643"],"award-info":[{"award-number":["UIDB\/00324\/2020","FCT\/MCTES","Teor\u00eda de Lie y Teor\u00eda de Espacios de Banach","FQM298","FEDER-UCA18-107643"]}]},{"name":"PAI","award":["UIDB\/00324\/2020","FCT\/MCTES","Teor\u00eda de Lie y Teor\u00eda de Espacios de Banach","FQM298","FEDER-UCA18-107643"],"award-info":[{"award-number":["UIDB\/00324\/2020","FCT\/MCTES","Teor\u00eda de Lie y Teor\u00eda de Espacios de Banach","FQM298","FEDER-UCA18-107643"]}]},{"name":"Spanish project \u2018Algebras no conmutativas y de caminos de Leavitt. Algebras de evoluci\u00f3n. Estructuras de Lie y variedades de Einstein\u2019","award":["UIDB\/00324\/2020","FCT\/MCTES","Teor\u00eda de Lie y Teor\u00eda de Espacios de Banach","FQM298","FEDER-UCA18-107643"],"award-info":[{"award-number":["UIDB\/00324\/2020","FCT\/MCTES","Teor\u00eda de Lie y Teor\u00eda de Espacios de Banach","FQM298","FEDER-UCA18-107643"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Mathematics"],"abstract":"<jats:p>In this work, we study a linear operator f on a pre-Euclidean space V by using properties of a corresponding graph. Given a basis B of V, we present a decomposition of V as an orthogonal direct sum of certain linear subspaces {Ui}i\u2208I, each one admitting a basis inherited from B, in such way that f=\u2211i\u2208Ifi. Each fi is a linear operator satisfying certain conditions with respect to Ui. Considering this new hypothesis, we assure the existence of an isomorphism between the graphs of f relative to two different bases. We also study the minimality of V by using the graph of f relative to B.<\/jats:p>","DOI":"10.3390\/math11030725","type":"journal-article","created":{"date-parts":[[2023,2,1]],"date-time":"2023-02-01T05:04:48Z","timestamp":1675227888000},"page":"725","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["Decomposition of Linear Operators on Pre-Euclidean Spaces by Means of Graphs"],"prefix":"10.3390","volume":"11","author":[{"given":"Hani","family":"Abdelwahab","sequence":"first","affiliation":[{"name":"Department of Mathematics, Faculty of Science, Mansoura University, Mansoura 35516, Egypt"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-1369-3737","authenticated-orcid":false,"given":"Elisabete","family":"Barreiro","sequence":"additional","affiliation":[{"name":"University of Coimbra, CMUC, Department of Mathematics, FCTUC, Largo D. Dinis, 3000-143 Coimbra, Portugal"}]},{"given":"Antonio J.","family":"Calder\u00f3n","sequence":"additional","affiliation":[{"name":"Department of Mathematics, University of C\u00e1diz, 11510 Puerto Real, Spain"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-4842-6735","authenticated-orcid":false,"given":"Jos\u00e9 M.","family":"S\u00e1nchez","sequence":"additional","affiliation":[{"name":"Department of Mathematics, University of C\u00e1diz, 11510 Puerto Real, Spain"}]}],"member":"1968","published-online":{"date-parts":[[2023,2,1]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","unstructured":"Barreiro, E., Calder\u00f3n, A.J., Lopes, S.A., and S\u00e1nchez, J.M. (2022). Leibniz algebras and graphs. Linear Multilinear Algebra.","DOI":"10.1080\/03081087.2022.2092048"},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"69","DOI":"10.1016\/j.dam.2018.03.020","article-title":"Bilinear maps and graphs","volume":"263","author":"Navarro","year":"2019","journal-title":"Discrete Appl. Math."},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"979","DOI":"10.1007\/s10801-021-01034-w","article-title":"On quantum adjacency algebras of Doob graphs and their irreducible modules","volume":"54","author":"Morales","year":"2021","journal-title":"J. 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