{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,3,27]],"date-time":"2026-03-27T12:13:21Z","timestamp":1774613601722,"version":"3.50.1"},"reference-count":26,"publisher":"MDPI AG","issue":"9","license":[{"start":{"date-parts":[[2025,8,28]],"date-time":"2025-08-28T00:00:00Z","timestamp":1756339200000},"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>In this paper, we introduce the concept of the \u201cgraph primal,\u201d which serves as the dual structure to the \u201cgraph grill\u201d. We present several results associated with graph primal operations. Moreover, we introduce two new graph-local functions on graph adjacency topological spaces (graph ATSs). We then explore the basic properties of the proposed graph-local functions and describe a method for generating two new graph ATSs from existing ones via graph primals. In addition, we examine several fundamental properties and connections of the resulting topologies, supported by some counterexamples. Furthermore, we characterize the nature of the open sets of these new topologies in terms of closure operators. Finally, we assess the compatibility of graph ATSs with the graph primal concept.<\/jats:p>","DOI":"10.3390\/axioms14090662","type":"journal-article","created":{"date-parts":[[2025,8,28]],"date-time":"2025-08-28T15:03:14Z","timestamp":1756393394000},"page":"662","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":3,"title":["On Graph Primal Topological Spaces"],"prefix":"10.3390","volume":"14","author":[{"given":"Dali","family":"Shi","sequence":"first","affiliation":[{"name":"College of Accounting, Guangzhou College of Technology and Business, Guangzhou 528138, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Salah E.","family":"Abbas","sequence":"additional","affiliation":[{"name":"Mathematics Department, Faculty of Science, Sohag University, Sohag 82524, Egypt"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0009-0006-3726-9013","authenticated-orcid":false,"given":"Hossam M.","family":"Khiamy","sequence":"additional","affiliation":[{"name":"Mathematics Department, Faculty of Science, Sohag University, Sohag 82524, Egypt"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-9010-0237","authenticated-orcid":false,"given":"Ismail","family":"Ibedou","sequence":"additional","affiliation":[{"name":"Department of Mathematics, Faculty of Science, Benha University, Benha 13518, Egypt"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2025,8,28]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","unstructured":"Bondy, J.A., and Murty, U. 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Axioms, 14.","DOI":"10.3390\/axioms14020084"},{"key":"ref_5","first-page":"266","article-title":"Some properties of the topological spaces generated from the simple undirected graphs","volume":"41","author":"Sari","year":"2023","journal-title":"Sigma J. Eng. Nat. Sci."},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"400","DOI":"10.1007\/s40314-024-02910-4","article-title":"A new multi-ideal nano-topological model via neighborhoods for diagnosis and cure of dengue","volume":"43","author":"Kaur","year":"2024","journal-title":"Comp. Appl. Math."},{"key":"ref_7","first-page":"295","article-title":"On the computer enumeration of finite topologies","volume":"10","author":"Evans","year":"1967","journal-title":"Commun. Assoc. Comp. Mach."},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"105","DOI":"10.7146\/math.scand.a-10849","article-title":"The lattice-graph of the topology of a transitive directed graph","volume":"21","author":"Anderson","year":"1967","journal-title":"Math. Scand."},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"291","DOI":"10.4067\/S0716-09172010000300008","article-title":"Finite topology and digraphs","volume":"29","author":"Marijuan","year":"2010","journal-title":"Proyecciones"},{"key":"ref_10","first-page":"647","article-title":"An Alexandroff topology on graphs","volume":"39","author":"Amiri","year":"2013","journal-title":"Bull. Iran. Math. Soc."},{"key":"ref_11","first-page":"44","article-title":"Topological spaces associated with simple graphs","volume":"9","author":"Kilicman","year":"2018","journal-title":"J. Math. Anal."},{"key":"ref_12","first-page":"79","article-title":"On a finite topological space induced by Hop neighbourhoods of a graph","volume":"21","author":"Nianga","year":"2019","journal-title":"Adv. Appl. Dis. Math."},{"key":"ref_13","first-page":"232","article-title":"Topologies on the edges set of directed graphs","volume":"18","author":"Abdu","year":"2018","journal-title":"J. Math. Comp. Sci."},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"5801","DOI":"10.1002\/mma.4726","article-title":"New types of topological structures via graphs","volume":"41","author":"Nada","year":"2018","journal-title":"Math. Methods Appl. Sci."},{"key":"ref_15","first-page":"33","article-title":"Topologies induced on vertex set of graphs","volume":"22","author":"Lalithambigai","year":"2023","journal-title":"Mapana-J. Sci."},{"key":"ref_16","unstructured":"Kuratowski, K. (1966). Topology, Academic Press."},{"key":"ref_17","first-page":"595","article-title":"Theory of filters","volume":"205","author":"Cartan","year":"1937","journal-title":"C. R. Acad. Sci."},{"key":"ref_18","first-page":"171","article-title":"Sur les notions de filter et grille","volume":"224","author":"Choquet","year":"1947","journal-title":"C. R. Acad. Sci."},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"1","DOI":"10.5269\/bspm.66792","article-title":"Primal topological spaces","volume":"43","author":"Acharjee","year":"2025","journal-title":"Bol. Soc. Parana Mat."},{"key":"ref_20","doi-asserted-by":"crossref","unstructured":"Al-Saadi, H., and Al-Malki, H. (2024). Categories of open sets in generalized primal topological spaces. Mathematics, 12.","DOI":"10.3390\/math12020207"},{"key":"ref_21","doi-asserted-by":"crossref","unstructured":"Al-Omari, A., and Alqahtani, M.H. (2023). Primal structure with closure operators and their applications. Mathematics, 11.","DOI":"10.3390\/math11244946"},{"key":"ref_22","first-page":"151","article-title":"Primal proximity spaces","volume":"66","author":"Acharjee","year":"2024","journal-title":"Mathematica"},{"key":"ref_23","doi-asserted-by":"crossref","first-page":"10756","DOI":"10.3934\/math.2024525","article-title":"Some operators in soft primal spaces","volume":"9","author":"Alqahtani","year":"2024","journal-title":"AIMS Math."},{"key":"ref_24","first-page":"426","article-title":"Topologies induced by graph grills on vertex set of graphs","volume":"20","author":"Lalithambigai","year":"2022","journal-title":"Neuroquantology"},{"key":"ref_25","first-page":"1","article-title":"Topology via graph ideals","volume":"48","author":"Gnanchandra","year":"2023","journal-title":"Ratio Math."},{"key":"ref_26","doi-asserted-by":"crossref","unstructured":"Chartrand, G., Lesniak, L., and Zhang, P. (2016). Textbooks in Mathematics (Graphs and Di-Graphs), Taylor and Francis Group, LLC. 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