Exploring Modular Multiplicative Divisor Labeling with Minimum Spanning Tree Optimization using Kruskal’s Algorithm in Single Edge Deletion of the Join of Two Graphs

Exploring Modular Multiplicative Divisor Labeling with Minimum Spanning Tree Optimization using Kruskal’s Algorithm in Single Edge Deletion of the Join of Two Graphs | Research Square window.SnipcartSettings = { analytics: { enabled: false } }; (function() { var accessVector = localStorage.getItem('access_vector') || ''; window.dataLayer = window.dataLayer || []; if (accessVector) { window.dataLayer.push({ user: { profile: { profileInfo: { snid: accessVector } } } }); } })(); (function(w,d,s,l,i){w[l]=w[l]||[];w[l].push({'gtm.start':new Date().getTime(),event:'gtm.js'});var f=d.getElementsByTagName(s)[0],j=d.createElement(s),dl=l!='dataLayer'?'&l='+l:'';j.async=true;j.src='https://www.googletagmanager.com/gtm.js?id='+i+dl;f.parentNode.insertBefore(j,f);})(window,document,'script','dataLayer','GTM-K279D39R'); Browse Preprints In Review Journals COVID-19 Preprints AJE Video Bytes Research Tools Research Promotion AJE Professional Editing AJE Rubriq About Preprint Platform In Review Editorial Policies Our Team Advisory Board Help Center Sign In Submit a Preprint Cite Share Download PDF Research Article Exploring Modular Multiplicative Divisor Labeling with Minimum Spanning Tree Optimization using Kruskal’s Algorithm in Single Edge Deletion of the Join of Two Graphs P. Kalarani, R. Revathi, M. Sathiragavan, V. Kamalakannan, B. Kalaiselvi This is a preprint; it has not been peer reviewed by a journal. https://doi.org/ 10.21203/rs.3.rs-6714541/v1 This work is licensed under a CC BY 4.0 License Status: Posted Version 1 posted You are reading this latest preprint version Abstract This research investigates the integration of Modular Multiplicative Divisor (MMD) labeling with Minimum Spanning Tree (MST) analysis by using Kruskals algorithum, focusing on the influence of single edge deletions in the join of two graphs, denoted as (𝐾1+𝐾1,𝜆)−𝜎1𝜌𝜆. The article presents a unique paradigm for preserving MMD labeling following edge deletions while simplifying complex graph structures, guaranteeing crucial connectivity and labeling attributes are retained. This study extends our theoretical understanding of graph labeling and provides practical strategies for optimizing network design, particularly in cases that need structural alterations. The findings demonstrate the proposed method's robustness in maintaining network integrity, making it a powerful tool for both theoretical insights and practical applications in network optimization. MSC : 05C30, 05C70, 05C76, 05C78, 05C99, 68R10. Graph labeling MMD labeling join of two graphs Edge deletion graph Tree Spanning Tree and Minimum Spanning Tree Full Text Additional Declarations No competing interests reported. Cite Share Download PDF Status: Posted Version 1 posted You are reading this latest preprint version Research Square lets you share your work early, gain feedback from the community, and start making changes to your manuscript prior to peer review in a journal. As a division of Research Square Company, we’re committed to making research communication faster, fairer, and more useful. We do this by developing innovative software and high quality services for the global research community. Our growing team is made up of researchers and industry professionals working together to solve the most critical problems facing scientific publishing. Also discoverable on Platform About Our Team In Review Editorial Policies Advisory Board Help Center Resources Author Services Accessibility API Access RSS feed Manage Cookie Preferences © Research Square 2026 | ISSN 2693-5015 (online) Privacy Policy Terms of Service Do Not Sell My Personal Information {"props":{"pageProps":{"initialData":{"identity":"rs-6714541","acceptedTermsAndConditions":true,"allowDirectSubmit":true,"archivedVersions":[],"articleType":"Research Article","associatedPublications":[],"authors":[{"id":471311574,"identity":"1691fe13-95aa-48aa-95f7-fd4ae7e9da65","order_by":0,"name":"P. Kalarani","email":"","orcid":"","institution":"Saveetha Institute of Medical And Technical Sciences","correspondingAuthor":false,"prefix":"","firstName":"P.","middleName":"","lastName":"Kalarani","suffix":""},{"id":471311575,"identity":"41272967-b861-40ee-8ecd-a84912ef24cd","order_by":1,"name":"R. Revathi","email":"data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAZAAAAAyAQMAAABI0h/eAAAABlBMVEX///8AAABVwtN+AAAACXBIWXMAAA7EAAAOxAGVKw4bAAABD0lEQVRIiWNgGAWjYLACxgYJKOPfPzkQfeABIS0HYVoY2A4Yg7UkENYCY7EdSASz8Wkxb+8x/Pxxh0U+/+z2i58LeO6kzw87/BBoi52cbgN2LTJnzhhLHDwjYTnjzpli6RkSz3I33k4zAGpJNjY7gF2LhERagsTBNgkDhhs5CdI8Bsy5G2cngLQcSNyGS4v8s+QfIC3yN3KSf/MkMKcbzk7/gF+LBPMxsC0GN9KPSfMcOJwgL51DwBae5GMWZ4FaDG/ksFnzNqQZbpDOKTiQYIDHL+wHm29UttUZyN1If3ybt8FGXn52+uYPHyrs5HBpQQI8BmDKAKzSgKByEGB/AKbkG4hSPQpGwSgYBSMIAAClDGUGv83BjQAAAABJRU5ErkJggg==","orcid":"","institution":"Saveetha Institute of Medical And Technical Sciences","correspondingAuthor":true,"prefix":"","firstName":"R.","middleName":"","lastName":"Revathi","suffix":""},{"id":471311576,"identity":"f857df6a-62ac-4f5d-bfbc-81ca3ac8a5b2","order_by":2,"name":"M. Sathiragavan","email":"","orcid":"","institution":"Rajalakshmi Engineering College","correspondingAuthor":false,"prefix":"","firstName":"M.","middleName":"","lastName":"Sathiragavan","suffix":""},{"id":471311577,"identity":"3809a38d-461f-436f-b2df-aa59ce80b90f","order_by":3,"name":"V. Kamalakannan","email":"","orcid":"","institution":"Saveetha Engineering College","correspondingAuthor":false,"prefix":"","firstName":"V.","middleName":"","lastName":"Kamalakannan","suffix":""},{"id":471311578,"identity":"592aea68-c1fd-4ae4-a41d-45dc3b7a5240","order_by":4,"name":"B. Kalaiselvi","email":"","orcid":"","institution":"Tagore college of Arts and Science","correspondingAuthor":false,"prefix":"","firstName":"B.","middleName":"","lastName":"Kalaiselvi","suffix":""}],"badges":[],"createdAt":"2025-05-21 08:53:17","currentVersionCode":1,"declarations":"","doi":"10.21203/rs.3.rs-6714541/v1","doiUrl":"https://doi.org/10.21203/rs.3.rs-6714541/v1","draftVersion":[],"editorialEvents":[],"editorialNote":"","failedWorkflow":false,"files":[{"id":94984985,"identity":"a784a778-285c-4f20-af83-f6f5680d2679","added_by":"auto","created_at":"2025-11-03 06:57:07","extension":"pdf","order_by":1,"title":"","display":"","copyAsset":false,"role":"manuscript-pdf","size":820459,"visible":true,"origin":"","legend":"","description":"","filename":"operationRforumEdgedeletionjoinoftwographs.pdf","url":"https://assets-eu.researchsquare.com/files/rs-6714541/v1_covered_ba9a52d0-6a9e-4e9b-a100-80ff76e1615a.pdf"}],"financialInterests":"No competing interests reported.","formattedTitle":"Exploring Modular Multiplicative Divisor Labeling with Minimum Spanning Tree Optimization using Kruskal’s Algorithm in Single Edge Deletion of the Join of Two Graphs","fulltext":[],"fulltextSource":"","fullText":"","funders":[],"hasAdminPriorityOnWorkflow":false,"hasManuscriptDocX":false,"hasOptedInToPreprint":true,"hasPassedJournalQc":"","hasAnyPriority":false,"hideJournal":true,"highlight":"","institution":"","isAcceptedByJournal":false,"isAuthorSuppliedPdf":true,"isDeskRejected":"","isHiddenFromSearch":false,"isInQc":false,"isInWorkflow":false,"isPdf":true,"isPdfUpToDate":true,"isWithdrawnOrRetracted":false,"journal":{"display":true,"email":"[email protected]","identity":"researchsquare","isNatureJournal":false,"hasQc":true,"allowDirectSubmit":true,"externalIdentity":"","sideBox":"","snPcode":"","submissionUrl":"/submission","title":"Research Square","twitterHandle":"researchsquare","acdcEnabled":true,"dfaEnabled":false,"editorialSystem":"","reportingPortfolio":"","inReviewEnabled":false,"inReviewRevisionsEnabled":true},"keywords":"Graph labeling, MMD labeling, join of two graphs, Edge deletion graph, Tree, Spanning Tree and Minimum Spanning Tree","lastPublishedDoi":"10.21203/rs.3.rs-6714541/v1","lastPublishedDoiUrl":"https://doi.org/10.21203/rs.3.rs-6714541/v1","license":{"name":"CC BY 4.0","url":"https://creativecommons.org/licenses/by/4.0/"},"manuscriptAbstract":"\u003cp\u003eThis research investigates the integration of Modular Multiplicative Divisor (MMD) labeling with Minimum Spanning Tree (MST) analysis by using Kruskals algorithum, focusing on the influence of single edge deletions in the join of two graphs, denoted as (𝐾1+𝐾1,𝜆)−𝜎1𝜌𝜆. The \u0026nbsp;article presents a unique paradigm for preserving MMD labeling following edge deletions while simplifying complex graph structures, guaranteeing crucial connectivity and labeling attributes are retained. This study extends our theoretical understanding of graph labeling and provides practical strategies for optimizing network design, particularly in cases that need structural alterations. The findings demonstrate the proposed method's robustness in maintaining network integrity, making it a powerful tool for both theoretical insights and practical applications in network optimization.\u003c/p\u003e\n\u003cp\u003e\u003cstrong\u003eMSC\u003c/strong\u003e: 05C30, 05C70, 05C76, 05C78, 05C99, 68R10.\u003c/p\u003e","manuscriptTitle":"Exploring Modular Multiplicative Divisor Labeling with Minimum Spanning Tree Optimization using Kruskal’s Algorithm in Single Edge Deletion of the Join of Two Graphs","msid":"","msnumber":"","nonDraftVersions":[{"code":1,"date":"2025-06-16 06:46:22","doi":"10.21203/rs.3.rs-6714541/v1","editorialEvents":[{"type":"communityComments","content":0}],"status":"published","journal":{"display":true,"email":"[email protected]","identity":"researchsquare","isNatureJournal":false,"hasQc":true,"allowDirectSubmit":true,"externalIdentity":"","sideBox":"","snPcode":"","submissionUrl":"/submission","title":"Research Square","twitterHandle":"researchsquare","acdcEnabled":true,"dfaEnabled":false,"editorialSystem":"","reportingPortfolio":"","inReviewEnabled":false,"inReviewRevisionsEnabled":true}}],"origin":"","ownerIdentity":"19edf956-72bf-4125-97c9-bc1cd31df335","owner":[],"postedDate":"June 16th, 2025","published":true,"recentEditorialEvents":[],"rejectedJournal":[],"revision":"","amendment":"","status":"posted","subjectAreas":[],"tags":[],"updatedAt":"2025-10-31T10:09:02+00:00","versionOfRecord":[],"versionCreatedAt":"2025-06-16 06:46:22","video":"","vorDoi":"","vorDoiUrl":"","workflowStages":[]},"version":"v1","identity":"rs-6714541","journalConfig":"researchsquare"},"__N_SSP":true},"page":"/article/[identity]/[[...version]]","query":{"redirect":"/article/rs-6714541","identity":"rs-6714541","version":["v1"]},"buildId":"8U1c8b4HqxoKbykW_rLl7","isFallback":false,"isExperimentalCompile":false,"dynamicIds":[84888],"gssp":true,"scriptLoader":[]}