Sigma Bond Calculation via Euler’s Formulafor Planar and Polyhedral Molecular Graphs

Sigma Bond Calculation via Euler’s Formulafor Planar and Polyhedral Molecular 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 Sigma Bond Calculation via Euler’s Formulafor Planar and Polyhedral Molecular Graphs Arvind Gyandatt Mishra This is a preprint; it has not been peer reviewed by a journal. https://doi.org/ 10.21203/rs.3.rs-7340867/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 Sigma (σ) bonds form the fundamental backbone of molecular connectivity, repre-senting the first covalent bond between two atoms in single, double, or triple bonds.Determining the number of σ bonds in a molecule is a basic yet essential task in struc-tural chemistry, with implications for stability, reactivity, and computational modeling.Traditional approaches require either manual bond counting from chemical drawingsor computational analysis of 3D structures, both of which can be time-consuming andprone to error for large, complex, or unconventional molecules.In this work, we present a mathematically rigorous and computationally efficientformula that predicts the number of σ bonds from three simple topological parameters:the total number of atoms (T ), the number of rings or faces (R), and a polyhedral indi-cator (ϕ). Derived directly from Euler’s characteristic for molecular graphs, the formula: σ = T +R −ϕ Achieves perfect agreement with known values across a diverse validationset of 50 molecules, including simple organics, heterocycles, ions, biomolecules, poly-hedral clusters, and extended frameworks. We provide a complete proof, schematicillustrations for planar and spherical embeddings, and an algorithmic flowchart. Thesimplicity, accuracy, and broad applicability make it a powerful tool for chemical in-formatics, education, and rapid molecular screening. For connected genus-0 moleculargraphs. Limitations for higher-genus/disconnected systems are noted. Sigma bonds the Euler’s formula molecular topology graph theory chemical informatics 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-7340867","acceptedTermsAndConditions":true,"allowDirectSubmit":true,"archivedVersions":[],"articleType":"Research Article","associatedPublications":[],"authors":[{"id":502064365,"identity":"606dc8e7-2dbf-4c81-9ff3-eee33f5d9016","order_by":0,"name":"Arvind Gyandatt Mishra","email":"data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAZAAAAAyAQMAAABI0h/eAAAABlBMVEX///8AAABVwtN+AAAACXBIWXMAAA7EAAAOxAGVKw4bAAABCUlEQVRIiWNgGAWjYJCCAwxsDEDEA2RWgPiMDQiSsJYzRGoBqQcCoBbGNoQYTi38s88YHi4oY4jm4z977MPPeXVyuu2H2yR+MNjIbjiAXYvEuRyDwzPOMeS2SeQlz+zddtjY7Exim2QPQ5oxLi0MZ9gSDvO2gbTwGDPwbjuQCERANsPhRFxa5OFa+M8YM/6dU5e47fzDNsk/DP9xajE4w3wAooUhx5iZt4E5cduNxDZpHoYDOLUYgrTwnJMAOgyoReYY0C83HjZbyxgkG8/EoUXuDGPzZ54ym9z5/UCHvampkzM7n/7w5psKO9k+XN6HAAkUHosEgwFe5ZiA+QOJGkbBKBgFo2B4AwDKU18GdPNvTAAAAABJRU5ErkJggg==","orcid":"","institution":"","correspondingAuthor":true,"prefix":"","firstName":"Arvind","middleName":"Gyandatt","lastName":"Mishra","suffix":""}],"badges":[],"createdAt":"2025-08-10 21:08:10","currentVersionCode":1,"declarations":"","doi":"10.21203/rs.3.rs-7340867/v1","doiUrl":"https://doi.org/10.21203/rs.3.rs-7340867/v1","draftVersion":[],"editorialEvents":[],"editorialNote":"","failedWorkflow":false,"files":[{"id":89349205,"identity":"f66727bb-4448-41f2-8cdc-2bab3f82ca81","added_by":"auto","created_at":"2025-08-19 05:45:54","extension":"pdf","order_by":1,"title":"","display":"","copyAsset":false,"role":"manuscript-pdf","size":342966,"visible":true,"origin":"","legend":"","description":"","filename":"SigmaBondCalculation.pdf","url":"https://assets-eu.researchsquare.com/files/rs-7340867/v1_covered_8110c961-e01d-4071-8588-59367254b23e.pdf"}],"financialInterests":"No competing interests reported.","formattedTitle":"Sigma Bond Calculation via Euler’s Formulafor Planar and Polyhedral Molecular Graphs","fulltext":[],"fulltextSource":"","fullText":"","funders":[],"hasAdminPriorityOnWorkflow":false,"hasManuscriptDocX":false,"hasOptedInToPreprint":true,"hasPassedJournalQc":"","hasAnyPriority":true,"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":"Sigma bonds, the Euler’s formula, molecular topology, graph theory, chemical informatics","lastPublishedDoi":"10.21203/rs.3.rs-7340867/v1","lastPublishedDoiUrl":"https://doi.org/10.21203/rs.3.rs-7340867/v1","license":{"name":"CC BY 4.0","url":"https://creativecommons.org/licenses/by/4.0/"},"manuscriptAbstract":"Sigma (σ) bonds form the fundamental backbone of molecular connectivity, repre-senting the first covalent bond between two atoms in single, double, or triple bonds.Determining the number of σ bonds in a molecule is a basic yet essential task in struc-tural chemistry, with implications for stability, reactivity, and computational modeling.Traditional approaches require either manual bond counting from chemical drawingsor computational analysis of 3D structures, both of which can be time-consuming andprone to error for large, complex, or unconventional molecules.In this work, we present a mathematically rigorous and computationally efficientformula that predicts the number of σ bonds from three simple topological parameters:the total number of atoms (T ), the number of rings or faces (R), and a polyhedral indi-cator (ϕ). Derived directly from Euler’s characteristic for molecular graphs, the formula:\nσ = T +R −ϕ\nAchieves perfect agreement with known values across a diverse validationset of 50 molecules, including simple organics, heterocycles, ions, biomolecules, poly-hedral clusters, and extended frameworks. We provide a complete proof, schematicillustrations for planar and spherical embeddings, and an algorithmic flowchart. Thesimplicity, accuracy, and broad applicability make it a powerful tool for chemical in-formatics, education, and rapid molecular screening. For connected genus-0 moleculargraphs. 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