{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,10]],"date-time":"2025-10-10T01:18:16Z","timestamp":1760059096765,"version":"build-2065373602"},"reference-count":20,"publisher":"MDPI AG","issue":"5","license":[{"start":{"date-parts":[[2025,5,20]],"date-time":"2025-05-20T00:00:00Z","timestamp":1747699200000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"name":"Deanship of Research and Graduate Studies at King Khalid University","award":["RGP2\/339\/45"],"award-info":[{"award-number":["RGP2\/339\/45"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Axioms"],"abstract":"<jats:p>This paper extends the concept of the total graph T\u0393(R) associated with a commutative ring to the three-fold Cartesian product R=Zn\u00d7Zm\u00d7Zp, where n,m,p&gt;1. We present complete and self-contained proofs for a wide range of graph-theoretic properties of T\u0393(R), including connectivity, diameter, regularity conditions, clique and independence numbers, and exact criteria for Hamiltonicity and Eulericity. We also derive improved lower bounds for the genus and characterize the automorphism group in both general and symmetric cases. Each result is illustrated through concrete numerical examples for clarity. Beyond theoretical contributions, we discuss potential applications in cryptographic key-exchange systems, fault-tolerant network architectures, and algebraic code design. This work generalizes and deepens prior studies on two-factor total graphs, and establishes a foundational framework for future exploration of higher-dimensional total graphs over finite commutative rings.<\/jats:p>","DOI":"10.3390\/axioms14050386","type":"journal-article","created":{"date-parts":[[2025,5,20]],"date-time":"2025-05-20T09:33:47Z","timestamp":1747733627000},"page":"386","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["Exploring the Structural and Traversal Properties of Total Graphs over Finite Rings"],"prefix":"10.3390","volume":"14","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-4167-3119","authenticated-orcid":false,"given":"Ali","family":"Al Khabyah","sequence":"first","affiliation":[{"name":"Department of Mathematics, College of Science, King Khalid University, Abha 61413, Saudi Arabia"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-9169-8488","authenticated-orcid":false,"family":"Nazim","sequence":"additional","affiliation":[{"name":"Department of Applied Sciences, Meerut Institute of Engineering and Technology (MIET), Meerut 250005, India"}]},{"given":"Ikram","family":"Ali","sequence":"additional","affiliation":[{"name":"Department of Computer Science, School of Engineering & Technology, Shri Venkateshwara University, Gajraula, Amroha 244236, India"}]}],"member":"1968","published-online":{"date-parts":[[2025,5,20]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"208","DOI":"10.1016\/0021-8693(88)90202-5","article-title":"Coloring of commutative rings","volume":"116","author":"Beck","year":"1988","journal-title":"J. Algebra"},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"434","DOI":"10.1006\/jabr.1998.7840","article-title":"The zero-divisor graph of a commutative ring","volume":"217","author":"Anderson","year":"1999","journal-title":"J. Algebra"},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"174","DOI":"10.1016\/j.jalgebra.2006.01.019","article-title":"The diameter of a zero-divisor graph","volume":"301","author":"Lucas","year":"2006","journal-title":"J. Algebra"},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"1155","DOI":"10.1016\/j.disc.2006.07.025","article-title":"On zero-divisor graphs of small finite commutative rings","volume":"307","author":"Redmond","year":"2007","journal-title":"Discret. Math."},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"1551","DOI":"10.1216\/RMJ-2012-42-5-1551","article-title":"Rings whose total graphs have genus at most one","volume":"42","author":"Maimani","year":"2012","journal-title":"Rocky Mt. J. Math."},{"key":"ref_6","doi-asserted-by":"crossref","unstructured":"Rehman, N.U., and Alghamdi, A. (2023). On Normalized Laplacian Spectra of the Weakly Zero-Divisor Graph of the Ring Zn. Mathematics, 11.","DOI":"10.3390\/math11204310"},{"key":"ref_7","doi-asserted-by":"crossref","unstructured":"Rehman, N.U., Alghamdi, A., and Almotairi, E.S. (2024). Randi\u0107 spectrum of the weakly zero-divisor graph of the ring Zn. AKCE Int. J. Graphs Comb., 302\u2013309.","DOI":"10.1080\/09728600.2024.2358360"},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"515","DOI":"10.1007\/s12215-023-00927-y","article-title":"Exploring normalized distance Laplacian eigenvalues of the zero-divisor graph Zn","volume":"73","author":"Rehman","year":"2024","journal-title":"Rend. Circ. Mat. Palermo"},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"2706","DOI":"10.1016\/j.jalgebra.2008.06.028","article-title":"The total graph of a commutative ring","volume":"320","author":"Anderson","year":"2008","journal-title":"J. Algebra"},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1080\/09720529.2011.10698320","article-title":"A note on the total graph of Zn","volume":"14","author":"Chelvam","year":"2011","journal-title":"J. Discret. Math. Sci. Cryptogr."},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"3820","DOI":"10.1080\/00927872.2012.678956","article-title":"On the total graph and its complement of commutative rings","volume":"41","author":"Asir","year":"2013","journal-title":"Commun. Algebra"},{"key":"ref_12","first-page":"75","article-title":"A Study of the Total Graph","volume":"6","author":"Tehranian","year":"2011","journal-title":"Iran. J. Math. Sci. Inform."},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"4213","DOI":"10.37418\/amsj.9.6.103","article-title":"Total graph of regular graphs","volume":"9","author":"Athul","year":"2020","journal-title":"Adv. Math. Sci. J."},{"key":"ref_14","doi-asserted-by":"crossref","unstructured":"Gadge, P., and Joshi, V. (2024). Total graph of a lattice. Indian J. Pure Appl. Math.","DOI":"10.1007\/s13226-024-00551-1"},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"730","DOI":"10.1016\/j.akcej.2019.12.005","article-title":"Complement of the generalized total graph of fields","volume":"17","author":"Chelvam","year":"2020","journal-title":"AKCE Int. J. Graphs Comb."},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"75","DOI":"10.1016\/j.dam.2024.07.047","article-title":"Spectra of total graphs","volume":"359","author":"Bu","year":"2024","journal-title":"Discret. Appl. Math."},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"1550004","DOI":"10.1142\/S1793830915500044","article-title":"Total graph of the ring Zn\u00d7Zm","volume":"7","author":"Dhorajia","year":"2015","journal-title":"Discret. Math. Algorithms Appl."},{"key":"ref_18","first-page":"985","article-title":"Zero-divisor graphs of direct product of commutative rings","volume":"32","author":"Axtell","year":"2006","journal-title":"Houst. J. Math."},{"key":"ref_19","first-page":"71","article-title":"Critical graphs of given diameter","volume":"30","author":"Plesnik","year":"1975","journal-title":"Acta Fac. Rerum Nat. Univ. Comen. Math."},{"key":"ref_20","unstructured":"Chartrand, G., and Zhang, P. (2006). Introduction to Graph Theory, Tata McGraw-Hill."}],"container-title":["Axioms"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2075-1680\/14\/5\/386\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,9]],"date-time":"2025-10-09T17:35:52Z","timestamp":1760031352000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2075-1680\/14\/5\/386"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2025,5,20]]},"references-count":20,"journal-issue":{"issue":"5","published-online":{"date-parts":[[2025,5]]}},"alternative-id":["axioms14050386"],"URL":"https:\/\/doi.org\/10.3390\/axioms14050386","relation":{},"ISSN":["2075-1680"],"issn-type":[{"type":"electronic","value":"2075-1680"}],"subject":[],"published":{"date-parts":[[2025,5,20]]}}}