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<!DOCTYPE html> <html > <head> <meta charset="utf-8"> <meta rel="search" type="application/opensearchdescription+xml" href="/open_search.xml" title="Academia.edu"> <meta content="width=device-width, initial-scale=1" name="viewport"> <meta name="google-site-verification" content="bKJMBZA7E43xhDOopFZkssMMkBRjvYERV-NaN4R6mrs"> <meta name="csrf-param" content="authenticity_token" /> <meta name="csrf-token" content="hmw3YdUAyT1dWmDeLawoLqhUgoKHHhhcGUrYrn2jtEg419SOEXdUyMgf89n7mJ6fUfRrFNOw-NiFpaZ6z7zZOA" /> <meta name="citation_title" content="Remarks about the Arithmetic of Graphs" /> <meta name="citation_publication_date" content="2021/06/18" /> <meta name="citation_author" content="Oliver Knill" /> <meta name="twitter:card" content="summary" /> <meta name="twitter:url" content="https://www.academia.edu/125874970/Remarks_about_the_Arithmetic_of_Graphs" /> <meta name="twitter:title" content="Remarks about the Arithmetic of Graphs" /> <meta name="twitter:description" content="The arithmetic of N, Z, Q, R can be extended to a graph arithmetic where N is the semiring of finite simple graphs and where Z and Q are integral domains, culminating in a Banach algebra R. A single network completes to the Wiener algebra. We" /> <meta name="twitter:image" content="https://0.academia-photos.com/24636766/6662359/7528121/s200_oliver.knill.jpg" /> <meta property="fb:app_id" content="2369844204" /> <meta property="og:type" content="article" /> <meta property="og:url" content="https://www.academia.edu/125874970/Remarks_about_the_Arithmetic_of_Graphs" /> <meta property="og:title" content="Remarks about the Arithmetic of Graphs" /> <meta property="og:image" content="http://a.academia-assets.com/images/open-graph-icons/fb-paper.gif" /> <meta property="og:description" content="The arithmetic of N, Z, Q, R can be extended to a graph arithmetic where N is the semiring of finite simple graphs and where Z and Q are integral domains, culminating in a Banach algebra R. A single network completes to the Wiener algebra. We" /> <meta property="article:author" content="https://harvard.academia.edu/OliverKnill" /> <meta name="description" content="The arithmetic of N, Z, Q, R can be extended to a graph arithmetic where N is the semiring of finite simple graphs and where Z and Q are integral domains, culminating in a Banach algebra R. A single network completes to the Wiener algebra. We" /> <title>(PDF) Remarks about the Arithmetic of Graphs</title> <link rel="canonical" href="https://www.academia.edu/125874970/Remarks_about_the_Arithmetic_of_Graphs" /> <script async src="https://www.googletagmanager.com/gtag/js?id=G-5VKX33P2DS"></script> <script> window.dataLayer = window.dataLayer || []; function gtag(){dataLayer.push(arguments);} gtag('js', new Date()); gtag('config', 'G-5VKX33P2DS', { cookie_domain: 'academia.edu', send_page_view: false, }); gtag('event', 'page_view', { 'controller': "single_work", 'action': "show", 'controller_action': 'single_work#show', 'logged_in': 'false', 'edge': 'unknown', // Send nil if there is no A/B test bucket, in case some records get logged // with missing data - that way we can distinguish between the two cases. // ab_test_bucket should be of the form <ab_test_name>:<bucket> 'ab_test_bucket': null, }) </script> <script> var $controller_name = 'single_work'; var $action_name = "show"; var $rails_env = 'production'; var $app_rev = '1e60a92a442ff83025cbe4f252857ee7c49c0bbe'; var $domain = 'academia.edu'; var $app_host = "academia.edu"; var $asset_host = "academia-assets.com"; var $start_time = new Date().getTime(); var $recaptcha_key = "6LdxlRMTAAAAADnu_zyLhLg0YF9uACwz78shpjJB"; var $recaptcha_invisible_key = "6Lf3KHUUAAAAACggoMpmGJdQDtiyrjVlvGJ6BbAj"; var $disableClientRecordHit = false; </script> <script> window.require = { config: function() { return function() {} } } </script> <script> window.Aedu = window.Aedu || {}; window.Aedu.hit_data = null; window.Aedu.serverRenderTime = new Date(1740574633000); window.Aedu.timeDifference = new Date().getTime() - 1740574633000; </script> <script type="application/ld+json">{"@context":"https://schema.org","@type":"ScholarlyArticle","abstract":"The arithmetic of N, Z, Q, R can be extended to a graph arithmetic where N is the semiring of finite simple graphs and where Z and Q are integral domains, culminating in a Banach algebra R. A single network completes to the Wiener algebra. We illustrate the compatibility with topology and spectral theory. Multiplicative linear functionals like Euler characteristic, the Poincare polynomial or the zeta functions can be extended naturally. These functionals can also help with number theoretical questions. The story of primes is a bit different as the integers are not a unique factorization domain, because there are many additive primes. Most graphs are multiplicative primes.","author":[{"@context":"https://schema.org","@type":"Person","name":"Oliver Knill","url":"https://harvard.academia.edu/OliverKnill"}],"contributor":[],"dateCreated":"2024-11-26","dateModified":"2024-11-26","datePublished":"2021-06-18","headline":"Remarks about the Arithmetic of Graphs","image":"https://attachments.academia-assets.com/119841121/thumbnails/1.jpg","inLanguage":"en","keywords":["Mathematics","Semiring"],"publisher":{"@context":"https://schema.org","@type":"Organization","name":null},"sourceOrganization":[{"@context":"https://schema.org","@type":"EducationalOrganization","name":"harvard"}],"thumbnailUrl":"https://attachments.academia-assets.com/119841121/thumbnails/1.jpg","url":"https://www.academia.edu/125874970/Remarks_about_the_Arithmetic_of_Graphs"}</script><style type="text/css">@media(max-width: 567px){:root{--token-mode: Rebrand;--dropshadow: 0 2px 4px 0 #22223340;--primary-brand: #0645b1;--error-dark: #b60000;--success-dark: 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{"work":{"id":125874970,"created_at":"2024-11-26T23:03:16.052-08:00","from_world_paper_id":261374465,"updated_at":"2024-11-26T23:27:04.845-08:00","_data":{"abstract":"The arithmetic of N, Z, Q, R can be extended to a graph arithmetic where N is the semiring of finite simple graphs and where Z and Q are integral domains, culminating in a Banach algebra R. A single network completes to the Wiener algebra. We illustrate the compatibility with topology and spectral theory. Multiplicative linear functionals like Euler characteristic, the Poincare polynomial or the zeta functions can be extended naturally. These functionals can also help with number theoretical questions. The story of primes is a bit different as the integers are not a unique factorization domain, because there are many additive primes. Most graphs are multiplicative primes.","publication_date":"2021,6,18"},"document_type":"paper","pre_hit_view_count_baseline":null,"quality":"high","language":"en","title":"Remarks about the Arithmetic of Graphs","broadcastable":false,"draft":null,"has_indexable_attachment":true,"indexable":true}}["work"]; window.loswp.workCoauthors = [24636766]; window.loswp.locale = "en"; window.loswp.countryCode = "SG"; window.loswp.cwvAbTestBucket = ""; window.loswp.designVariant = "ds_vanilla"; window.loswp.fullPageMobileSutdModalVariant = "full_page_mobile_sutd_modal"; window.loswp.useOptimizedScribd4genScript = false; window.loginModal = {}; window.loginModal.appleClientId = 'edu.academia.applesignon'; window.userInChina = "false";</script><script defer="" src="https://accounts.google.com/gsi/client"></script><div class="ds-loswp-container"><div class="ds-work-card--grid-container"><div class="ds-work-card--container js-loswp-work-card"><div class="ds-work-card--cover"><div class="ds-work-cover--wrapper"><div class="ds-work-cover--container"><button class="ds-work-cover--clickable js-swp-download-button" data-signup-modal="{"location":"swp-splash-paper-cover","attachmentId":119841121,"attachmentType":"pdf"}"><img alt="First page of “Remarks about the Arithmetic of Graphs”" class="ds-work-cover--cover-thumbnail" src="https://0.academia-photos.com/attachment_thumbnails/119841121/mini_magick20241127-1-t8fg4k.png?1732691019" /><img alt="PDF Icon" class="ds-work-cover--file-icon" src="//a.academia-assets.com/images/single_work_splash/adobe_icon.svg" /><div class="ds-work-cover--hover-container"><span class="material-symbols-outlined" style="font-size: 20px" translate="no">download</span><p>Download Free PDF</p></div><div class="ds-work-cover--ribbon-container">Download Free PDF</div><div class="ds-work-cover--ribbon-triangle"></div></button></div></div></div><div class="ds-work-card--work-information"><h1 class="ds-work-card--work-title">Remarks about the Arithmetic of Graphs</h1><div class="ds-work-card--work-authors ds-work-card--detail"><a class="ds-work-card--author js-wsj-grid-card-author ds2-5-body-md ds2-5-body-link" data-author-id="24636766" href="https://harvard.academia.edu/OliverKnill"><img alt="Profile image of Oliver Knill" class="ds-work-card--author-avatar" src="https://0.academia-photos.com/24636766/6662359/7528121/s65_oliver.knill.jpg" />Oliver Knill</a></div><div class="ds-work-card--detail"><p class="ds-work-card--detail ds2-5-body-sm">2021</p><div class="ds-work-card--work-metadata"><div class="ds-work-card--work-metadata__stat"><span class="material-symbols-outlined" style="font-size: 20px" translate="no">visibility</span><p class="ds2-5-body-sm" id="work-metadata-view-count">…</p></div><div class="ds-work-card--work-metadata__stat"><span class="material-symbols-outlined" style="font-size: 20px" translate="no">description</span><p class="ds2-5-body-sm">16 pages</p></div><div class="ds-work-card--work-metadata__stat"><span class="material-symbols-outlined" style="font-size: 20px" translate="no">link</span><p class="ds2-5-body-sm">1 file</p></div></div><script>(async () => { const workId = 125874970; const worksViewsPath = "/v0/works/views?subdomain_param=api&work_ids%5B%5D=125874970"; const getWorkViews = async (workId) => { const response = await fetch(worksViewsPath); if (!response.ok) { throw new Error('Failed to load work views'); } const data = await response.json(); return data.views[workId]; }; // Get the view count for the work - we send this immediately rather than waiting for // the DOM to load, so it can be available as soon as possible (but without holding up // the backend or other resource requests, because it's a bit expensive and not critical). const viewCount = await getWorkViews(workId); const updateViewCount = (viewCount) => { try { const viewCountNumber = parseInt(viewCount, 10); if (viewCountNumber === 0) { // Remove the whole views element if there are zero views. document.getElementById('work-metadata-view-count')?.parentNode?.remove(); return; } const commaizedViewCount = viewCountNumber.toLocaleString(); const viewCountBody = document.getElementById('work-metadata-view-count'); if (!viewCountBody) { throw new Error('Failed to find work views element'); } viewCountBody.textContent = `${commaizedViewCount} views`; } catch (error) { // Remove the whole views element if there was some issue parsing. document.getElementById('work-metadata-view-count')?.parentNode?.remove(); throw new Error(`Failed to parse view count: ${viewCount}`, error); } }; // If the DOM is still loading, wait for it to be ready before updating the view count. if (document.readyState === "loading") { document.addEventListener('DOMContentLoaded', () => { updateViewCount(viewCount); }); // Otherwise, just update it immediately. } else { updateViewCount(viewCount); } })();</script></div><p class="ds-work-card--work-abstract ds-work-card--detail ds2-5-body-md">The arithmetic of N, Z, Q, R can be extended to a graph arithmetic where N is the semiring of finite simple graphs and where Z and Q are integral domains, culminating in a Banach algebra R. A single network completes to the Wiener algebra. We illustrate the compatibility with topology and spectral theory. Multiplicative linear functionals like Euler characteristic, the Poincare polynomial or the zeta functions can be extended naturally. These functionals can also help with number theoretical questions. The story of primes is a bit different as the integers are not a unique factorization domain, because there are many additive primes. Most graphs are multiplicative primes.</p><div class="ds-work-card--button-container"><button class="ds2-5-button js-swp-download-button" data-signup-modal="{"location":"continue-reading-button--work-card","attachmentId":119841121,"attachmentType":"pdf","workUrl":"https://www.academia.edu/125874970/Remarks_about_the_Arithmetic_of_Graphs"}">See full PDF</button><button class="ds2-5-button ds2-5-button--secondary js-swp-download-button" data-signup-modal="{"location":"download-pdf-button--work-card","attachmentId":119841121,"attachmentType":"pdf","workUrl":"https://www.academia.edu/125874970/Remarks_about_the_Arithmetic_of_Graphs"}"><span class="material-symbols-outlined" style="font-size: 20px" translate="no">download</span>Download PDF</button></div><div class="ds-signup-banner-trigger-container"><div class="ds-signup-banner-trigger ds-signup-banner-trigger-control"></div></div><div class="ds-signup-banner ds-signup-banner-control"><div id="ds-signup-banner-close-button"><button class="ds2-5-button ds2-5-button--secondary ds2-5-button--inverse"><span class="material-symbols-outlined" style="font-size: 20px" translate="no">close</span></button></div><div class="ds-signup-banner-ctas" data-impression-entity-id="125874970" data-impression-entity-type="2" data-impression-source="signup-banner"><img src="//a.academia-assets.com/images/academia-logo-capital-white.svg" /><h4 class="ds2-5-heading-serif-sm">Sign up for access to the world's latest research</h4><button class="ds2-5-button ds2-5-button--inverse ds2-5-button--full-width js-swp-download-button" data-signup-modal="{"location":"signup-banner"}">Sign up for free<span class="material-symbols-outlined" style="font-size: 20px" translate="no">arrow_forward</span></button></div><div class="ds-signup-banner-divider"></div><div class="ds-signup-banner-reasons"><div class="ds-signup-banner-reasons-item"><span class="material-symbols-outlined" style="font-size: 24px" translate="no">check</span><span>Get notified about relevant papers</span></div><div class="ds-signup-banner-reasons-item"><span class="material-symbols-outlined" style="font-size: 24px" translate="no">check</span><span>Save papers to use in your research</span></div><div class="ds-signup-banner-reasons-item"><span class="material-symbols-outlined" style="font-size: 24px" translate="no">check</span><span>Join the discussion with peers</span></div><div class="ds-signup-banner-reasons-item"><span class="material-symbols-outlined" style="font-size: 24px" translate="no">check</span><span>Track your impact</span></div></div></div><script>(() => { // Set up signup banner show/hide behavior: // 1. 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We know that because by the graph complement operation it is isomorphic to the strong Sabidussi ring with disjoint union as addition. We prove that the Euler characteristic is a ring homomorphism from the strong ring to the integers by demonstrating that the strong ring is homotopic to a Stanley-Reisner Cartesian ring. More generally, the Kuenneth formula holds on the strong ring so that the Poincaré polynomial is compatible with the ring structure. The Zykov ring has the clique number as a ring homomorphism. Furthermore, the Cartesian ring has the property that the functor which attaches to a graph the spectrum of its connection Laplacian is multiplicative. The reason is that the connection Laplacians do tensor under multiplication, similarly to what the adjacency matrix does for the weak ring. The strong ring product of two graphs contains both the weak and direct product graphs as subgraphs. The Zykov, Sabidussi or Stanley-Reisner rings are so manifestations of a network arithmetic which has remarkable cohomological properties, dimension and spectral compatibility but where arithmetic questions like the complexity of detecting primes or factoring are not yet studied well. We illustrate the Zykov arithmetic with examples, especially from the subring generated by point graphs which contains spheres, stars or complete bipartite graphs. While things are formulated in the language of graph theory, all constructions generalize to the larger category of finite abstract simplicial complexes.</p><div class="ds-related-work--ctas"><button class="ds2-5-text-link ds2-5-text-link--inline js-swp-download-button" data-signup-modal="{"location":"wsj-grid-card-download-pdf-modal","work_title":"On the arithmetic of graphs","attachmentId":118801357,"attachmentType":"pdf","work_url":"https://www.academia.edu/124606331/On_the_arithmetic_of_graphs","alternativeTracking":true}"><span class="material-symbols-outlined" style="font-size: 18px" translate="no">download</span><span class="ds2-5-text-link__content">Download free PDF</span></button><a class="ds2-5-text-link ds2-5-text-link--inline js-wsj-grid-card-view-pdf" href="https://www.academia.edu/124606331/On_the_arithmetic_of_graphs"><span class="ds2-5-text-link__content">View PDF</span><span class="material-symbols-outlined" style="font-size: 18px" translate="no">chevron_right</span></a></div></div><div class="ds-related-work--container js-wsj-grid-card" data-collection-position="1" data-entity-id="82373772" data-sort-order="default"><a class="ds-related-work--title js-wsj-grid-card-title ds2-5-body-md ds2-5-body-link" href="https://www.academia.edu/82373772/On_a_Connection_of_Number_Theory_with_Graph_Theory">On a Connection of Number Theory with Graph Theory</a><div class="ds-related-work--metadata"><a class="js-wsj-grid-card-author ds2-5-body-sm ds2-5-body-link" data-author-id="5365968" href="https://cvut.academia.edu/Alena%C5%A0olcov%C3%A1">Alena Šolcová</a></div><p class="ds-related-work--metadata ds2-5-body-xs">Czechoslovak Mathematical Journal, 2004</p><p class="ds-related-work--abstract ds2-5-body-sm">Institute of Mathematics of the Czech Academy of Sciences provides access to digitized documents strictly for personal use. Each copy of any part of this document must contain these Terms of use.</p><div class="ds-related-work--ctas"><button class="ds2-5-text-link ds2-5-text-link--inline js-swp-download-button" data-signup-modal="{"location":"wsj-grid-card-download-pdf-modal","work_title":"On a Connection of Number Theory with Graph Theory","attachmentId":88103927,"attachmentType":"pdf","work_url":"https://www.academia.edu/82373772/On_a_Connection_of_Number_Theory_with_Graph_Theory","alternativeTracking":true}"><span class="material-symbols-outlined" style="font-size: 18px" translate="no">download</span><span class="ds2-5-text-link__content">Download free PDF</span></button><a class="ds2-5-text-link ds2-5-text-link--inline js-wsj-grid-card-view-pdf" href="https://www.academia.edu/82373772/On_a_Connection_of_Number_Theory_with_Graph_Theory"><span class="ds2-5-text-link__content">View PDF</span><span class="material-symbols-outlined" style="font-size: 18px" translate="no">chevron_right</span></a></div></div><div class="ds-related-work--container js-wsj-grid-card" data-collection-position="2" data-entity-id="127716878" data-sort-order="default"><a class="ds-related-work--title js-wsj-grid-card-title ds2-5-body-md ds2-5-body-link" href="https://www.academia.edu/127716878/Algebraic_Graph_Theory">Algebraic Graph Theory</a><div class="ds-related-work--metadata"><a class="js-wsj-grid-card-author ds2-5-body-sm ds2-5-body-link" data-author-id="59887885" href="https://independent.academia.edu/CGodsil">Chris Godsil</a></div><p class="ds-related-work--metadata ds2-5-body-xs">Graduate texts in mathematics, 2001</p><p class="ds-related-work--abstract ds2-5-body-sm">TAKEUTIIZARING. Introduction to 35 ALEXANDERIWERMER. Several Complex Axiomatic Set Theory. 2nd ed. Variables and Banach Algebras. 3rd ed. 2 OXTOBY. Measure and Category. 2nd ed. 36 KELLEy/NAMIOKA et al. Linear 3 SCHAEFER. Topological Vector Spaces. Topological Spaces. 2nd ed. 37 MONK. Mathematical Logic. 4 HILTON/STAMMBACH. A Course in 38 GRAUERTIFRlTZSCHE. Several Complex Homological Algebra. 2nd ed. Variables. 5 MAC LANE. Categories for the Working 39 ARVESON. An Invitation to C*-Algebras. Mathematician. 2nd ed. 40 KEMENY/SNELLIKNAPP. Denumerable 6 HUGHES/PIPER. Projective Planes. Markov Chains. 2nd ed. 7 SERRE. A Course in Arithmetic. 41 ApOSTOL. Modular Functions and 8 TAKEUTIIZARING. Axiomatic Set Theory. Dirichlet Series in Number Theory. 9 HUMPHREYs. Introduction to Lie Algebras 2nd ed. and Representation Theory. 42 SERRE. Linear Representations of Finite COHEN. A Course in Simple Homotopy Groups. Theory. 43 GILLMAN/JERISON. Rings of Continuous CONWAY. Functions of One Complex Functions. Variable I. 2nd ed. 44 KENDIG. Elementary Algebraic Geometry. BEALS. Advanced Mathematical Analysis. 45 LoiNE. Probability Theory I. 4th ed. ANDERSONIFULLER. Rings and Categories 46 LOEVE. Probability Theory II. 4th ed. of Modules. 2nd ed.</p><div class="ds-related-work--ctas"><button class="ds2-5-text-link ds2-5-text-link--inline js-swp-download-button" data-signup-modal="{"location":"wsj-grid-card-download-pdf-modal","work_title":"Algebraic Graph Theory","attachmentId":121408713,"attachmentType":"pdf","work_url":"https://www.academia.edu/127716878/Algebraic_Graph_Theory","alternativeTracking":true}"><span class="material-symbols-outlined" style="font-size: 18px" translate="no">download</span><span class="ds2-5-text-link__content">Download free PDF</span></button><a class="ds2-5-text-link ds2-5-text-link--inline js-wsj-grid-card-view-pdf" href="https://www.academia.edu/127716878/Algebraic_Graph_Theory"><span class="ds2-5-text-link__content">View PDF</span><span class="material-symbols-outlined" style="font-size: 18px" translate="no">chevron_right</span></a></div></div><div class="ds-related-work--container js-wsj-grid-card" data-collection-position="3" data-entity-id="53626601" data-sort-order="default"><a class="ds-related-work--title js-wsj-grid-card-title ds2-5-body-md ds2-5-body-link" href="https://www.academia.edu/53626601/Some_Algebraic_Properties_of_a_Class_of_Integral_Graphs_Determined_by_Their_Spectrum">Some Algebraic Properties of a Class of Integral Graphs Determined by Their Spectrum</a><div class="ds-related-work--metadata"><a class="js-wsj-grid-card-author ds2-5-body-sm ds2-5-body-link" data-author-id="161064215" href="https://independent.academia.edu/alizafari5">ali zafari</a></div><p class="ds-related-work--metadata ds2-5-body-xs">Journal of Mathematics</p><p class="ds-related-work--abstract ds2-5-body-sm">Let Γ = V , E be a graph. If all the eigenvalues of the adjacency matrix of the graph Γ are integers, then we say that Γ is an integral graph. A graph Γ is determined by its spectrum if every graph cospectral to it is in fact isomorphic to it. In this paper, we investigate some algebraic properties of the Cayley graph Γ = Cay ℤ n , S , where n = p m ( p is a prime integer and m ∈ ℕ ) and S = a ∈ ℤ n | a , n = 1 . First, we show that Γ is an integral graph. Also, we determine the automorphism group of Γ . Moreover, we show that Γ and K v ▽ Γ are determined by their spectrum.</p><div class="ds-related-work--ctas"><button class="ds2-5-text-link ds2-5-text-link--inline js-swp-download-button" data-signup-modal="{"location":"wsj-grid-card-download-pdf-modal","work_title":"Some Algebraic Properties of a Class of Integral Graphs Determined by Their Spectrum","attachmentId":70383666,"attachmentType":"pdf","work_url":"https://www.academia.edu/53626601/Some_Algebraic_Properties_of_a_Class_of_Integral_Graphs_Determined_by_Their_Spectrum","alternativeTracking":true}"><span class="material-symbols-outlined" style="font-size: 18px" translate="no">download</span><span class="ds2-5-text-link__content">Download free PDF</span></button><a class="ds2-5-text-link ds2-5-text-link--inline js-wsj-grid-card-view-pdf" href="https://www.academia.edu/53626601/Some_Algebraic_Properties_of_a_Class_of_Integral_Graphs_Determined_by_Their_Spectrum"><span class="ds2-5-text-link__content">View PDF</span><span class="material-symbols-outlined" style="font-size: 18px" translate="no">chevron_right</span></a></div></div><div class="ds-related-work--container js-wsj-grid-card" data-collection-position="4" data-entity-id="4978153" data-sort-order="default"><a class="ds-related-work--title js-wsj-grid-card-title ds2-5-body-md ds2-5-body-link" href="https://www.academia.edu/4978153/Integer_Magic_Spectra_of_Functional_Extensions_of_Graphs">Integer-Magic Spectra of Functional Extensions of Graphs</a><div class="ds-related-work--metadata"><a class="js-wsj-grid-card-author ds2-5-body-sm ds2-5-body-link" data-author-id="6576158" href="https://independent.academia.edu/SinMinLee">Sin-Min Lee</a></div><p class="ds-related-work--abstract ds2-5-body-sm">For any kEN, a graph G = (V, E) is said to be ;:z k-magic if there exists a labeling Z: E( G) --+ ;:z k -{OJ such that the induced vertex set labeling Z+: V (G) --+ ;:z k defined by</p><div class="ds-related-work--ctas"><button class="ds2-5-text-link ds2-5-text-link--inline js-swp-download-button" data-signup-modal="{"location":"wsj-grid-card-download-pdf-modal","work_title":"Integer-Magic Spectra of Functional Extensions of Graphs","attachmentId":32225597,"attachmentType":"pdf","work_url":"https://www.academia.edu/4978153/Integer_Magic_Spectra_of_Functional_Extensions_of_Graphs","alternativeTracking":true}"><span class="material-symbols-outlined" style="font-size: 18px" translate="no">download</span><span class="ds2-5-text-link__content">Download free PDF</span></button><a class="ds2-5-text-link ds2-5-text-link--inline js-wsj-grid-card-view-pdf" href="https://www.academia.edu/4978153/Integer_Magic_Spectra_of_Functional_Extensions_of_Graphs"><span class="ds2-5-text-link__content">View PDF</span><span class="material-symbols-outlined" style="font-size: 18px" translate="no">chevron_right</span></a></div></div><div class="ds-related-work--container js-wsj-grid-card" data-collection-position="5" data-entity-id="16887935" data-sort-order="default"><a class="ds-related-work--title js-wsj-grid-card-title ds2-5-body-md ds2-5-body-link" href="https://www.academia.edu/16887935/Arithmetic_Properties_of_Eigenvalues_of_Generalized_Harper_Operators_on_Graphs">Arithmetic Properties of Eigenvalues of Generalized Harper Operators on Graphs</a><div class="ds-related-work--metadata"><a class="js-wsj-grid-card-author ds2-5-body-sm ds2-5-body-link" data-author-id="36367941" href="https://gc-cuny.academia.edu/J%C3%B3zefDodziuk">Józef Dodziuk</a></div><p class="ds-related-work--metadata ds2-5-body-xs">Communications in Mathematical Physics, 2006</p><p class="ds-related-work--abstract ds2-5-body-sm">Let Q denote the field of algebraic numbers in C. A discrete group G is said to have the σ-multiplier algebraic eigenvalue property, if for every matrix A ∈ M d (Q(G, σ)), regarded as an operator on l 2 (G) d , the eigenvalues of A are algebraic numbers, where σ ∈ Z 2 (G, U(Q)) is an algebraic multiplier, and U(Q) denotes the unitary elements of Q. Such operators include the Harper operator and the discrete magnetic Laplacian that occur in solid state physics. We prove that any finitely generated amenable, free or surface group has this property for any algebraic multiplier σ. In the special case when σ is rational (σ n =1 for some positive integer n) this property holds for a larger class of groups K containing free groups and amenable groups, and closed under taking directed unions and extensions with amenable quotients. Included in the paper are proofs of other spectral properties of such operators.</p><div class="ds-related-work--ctas"><button class="ds2-5-text-link ds2-5-text-link--inline js-swp-download-button" data-signup-modal="{"location":"wsj-grid-card-download-pdf-modal","work_title":"Arithmetic Properties of Eigenvalues of Generalized Harper Operators on Graphs","attachmentId":39239678,"attachmentType":"pdf","work_url":"https://www.academia.edu/16887935/Arithmetic_Properties_of_Eigenvalues_of_Generalized_Harper_Operators_on_Graphs","alternativeTracking":true}"><span class="material-symbols-outlined" style="font-size: 18px" translate="no">download</span><span class="ds2-5-text-link__content">Download free PDF</span></button><a class="ds2-5-text-link ds2-5-text-link--inline js-wsj-grid-card-view-pdf" href="https://www.academia.edu/16887935/Arithmetic_Properties_of_Eigenvalues_of_Generalized_Harper_Operators_on_Graphs"><span class="ds2-5-text-link__content">View PDF</span><span class="material-symbols-outlined" style="font-size: 18px" translate="no">chevron_right</span></a></div></div><div class="ds-related-work--container js-wsj-grid-card" data-collection-position="6" data-entity-id="8380639" data-sort-order="default"><a class="ds-related-work--title js-wsj-grid-card-title ds2-5-body-md ds2-5-body-link" href="https://www.academia.edu/8380639/Integer_Magic_Spectra_of_Functional_Extension_of_Graphs">Integer-Magic Spectra of Functional Extension of Graphs</a><div class="ds-related-work--metadata"><a class="js-wsj-grid-card-author ds2-5-body-sm ds2-5-body-link" data-author-id="16872404" href="https://sjsu.academia.edu/SinMinLee">Sin-Min Lee</a></div><p class="ds-related-work--abstract ds2-5-body-sm">For any kEN, a graph G = (V, E) is said to be ;:z k-magic if there exists a labeling Z: E( G) --+ ;:z k -{OJ such that the induced vertex set labeling Z+: V (G) --+ ;:z k defined by</p><div class="ds-related-work--ctas"><button class="ds2-5-text-link ds2-5-text-link--inline js-swp-download-button" data-signup-modal="{"location":"wsj-grid-card-download-pdf-modal","work_title":"Integer-Magic Spectra of Functional Extension of Graphs","attachmentId":48114810,"attachmentType":"pdf","work_url":"https://www.academia.edu/8380639/Integer_Magic_Spectra_of_Functional_Extension_of_Graphs","alternativeTracking":true}"><span class="material-symbols-outlined" style="font-size: 18px" translate="no">download</span><span class="ds2-5-text-link__content">Download free PDF</span></button><a class="ds2-5-text-link ds2-5-text-link--inline js-wsj-grid-card-view-pdf" href="https://www.academia.edu/8380639/Integer_Magic_Spectra_of_Functional_Extension_of_Graphs"><span class="ds2-5-text-link__content">View PDF</span><span class="material-symbols-outlined" style="font-size: 18px" translate="no">chevron_right</span></a></div></div><div class="ds-related-work--container js-wsj-grid-card" data-collection-position="7" data-entity-id="124606346" data-sort-order="default"><a class="ds-related-work--title js-wsj-grid-card-title ds2-5-body-md ds2-5-body-link" href="https://www.academia.edu/124606346/More_on_Numbers_and_Graphs">More on Numbers and Graphs</a><div class="ds-related-work--metadata"><a class="js-wsj-grid-card-author ds2-5-body-sm ds2-5-body-link" data-author-id="99076177" href="https://harvard.academia.edu/OKnill">Oliver Knill</a></div><p class="ds-related-work--metadata ds2-5-body-xs">arXiv (Cornell University), 2019</p><p class="ds-related-work--abstract ds2-5-body-sm">In this note we revisit a "ring of graphs" Q in which the set of finite simple graphs N extend the role of the natural numbers N and the signed graphs Z extend the role of the integers Z. We point out the existence of a norm which allows to complete Q to a real or complex Banach algebra R or C.</p><div class="ds-related-work--ctas"><button class="ds2-5-text-link ds2-5-text-link--inline js-swp-download-button" data-signup-modal="{"location":"wsj-grid-card-download-pdf-modal","work_title":"More on Numbers and Graphs","attachmentId":118801374,"attachmentType":"pdf","work_url":"https://www.academia.edu/124606346/More_on_Numbers_and_Graphs","alternativeTracking":true}"><span class="material-symbols-outlined" style="font-size: 18px" translate="no">download</span><span class="ds2-5-text-link__content">Download free PDF</span></button><a class="ds2-5-text-link ds2-5-text-link--inline js-wsj-grid-card-view-pdf" href="https://www.academia.edu/124606346/More_on_Numbers_and_Graphs"><span class="ds2-5-text-link__content">View PDF</span><span class="material-symbols-outlined" style="font-size: 18px" translate="no">chevron_right</span></a></div></div><div class="ds-related-work--container js-wsj-grid-card" data-collection-position="8" data-entity-id="73511164" data-sort-order="default"><a class="ds-related-work--title js-wsj-grid-card-title ds2-5-body-md ds2-5-body-link" href="https://www.academia.edu/73511164/Tropical_Arithmetics_and_Dot_Product_Representations_of_Graphs">Tropical Arithmetics and Dot Product Representations of Graphs</a><div class="ds-related-work--metadata"><a class="js-wsj-grid-card-author ds2-5-body-sm ds2-5-body-link" data-author-id="189053380" href="https://independent.academia.edu/NicoleTurner39">Nicole Turner</a></div><p class="ds-related-work--metadata ds2-5-body-xs">2015</p><p class="ds-related-work--abstract ds2-5-body-sm">A dot product representation (DPR) of a graph is a function that maps each vertex to a vector and two vertices are adjacent if and only if the dot product of their function values is greater than a given threshold. A tropical algebra is the antinegative semiring on IR∪{∞, −∞} with either min{a, b} replacing a+b and a+b replacing a•b (min-plus), or max{a, b} replacing a + b and a + b replacing a • b (max-plus), and the symbol ∞ is the additive identity in min-plus while −∞ is the additive identity in max-plus; the multiplicative identity is 0 in min-plus and in max-plus. Recall the threshold dimension of graph G is the minimum number of threshold graphs whose union is G. We study DPRs in the context of tropical semi-rings, and discuss results on minimizing the dimension of the space from which vectors must come in order to represent certain classes of graphs. These results differ depending on whether min-plus or maxplus is used, but a relationship is shown between the min-plus and max-plus results. Finally we show that max-plus dot product dimension and the threshold dimension of a graph are the same.</p><div class="ds-related-work--ctas"><button class="ds2-5-text-link ds2-5-text-link--inline js-swp-download-button" data-signup-modal="{"location":"wsj-grid-card-download-pdf-modal","work_title":"Tropical Arithmetics and Dot Product Representations of Graphs","attachmentId":82006070,"attachmentType":"pdf","work_url":"https://www.academia.edu/73511164/Tropical_Arithmetics_and_Dot_Product_Representations_of_Graphs","alternativeTracking":true}"><span class="material-symbols-outlined" style="font-size: 18px" translate="no">download</span><span class="ds2-5-text-link__content">Download free PDF</span></button><a class="ds2-5-text-link ds2-5-text-link--inline js-wsj-grid-card-view-pdf" href="https://www.academia.edu/73511164/Tropical_Arithmetics_and_Dot_Product_Representations_of_Graphs"><span class="ds2-5-text-link__content">View PDF</span><span class="material-symbols-outlined" style="font-size: 18px" translate="no">chevron_right</span></a></div></div><div class="ds-related-work--container js-wsj-grid-card" data-collection-position="9" data-entity-id="65667420" data-sort-order="default"><a class="ds-related-work--title js-wsj-grid-card-title ds2-5-body-md ds2-5-body-link" href="https://www.academia.edu/65667420/On_the_Generalized_Total_Graph_of_Fields_and_Its_Complement">On the Generalized Total Graph of Fields and Its Complement</a><div class="ds-related-work--metadata"><a class="js-wsj-grid-card-author ds2-5-body-sm ds2-5-body-link" data-author-id="191288031" href="https://independent.academia.edu/DrTTamizhChelvam">Dr. T. Tamizh Chelvam</a></div><p class="ds-related-work--metadata ds2-5-body-xs">2018</p><p class="ds-related-work--abstract ds2-5-body-sm">Let R be a commutative ring with identity, Z(R) its set of all zero-divisors, and H a nonempty proper multiplicative prime subset of R. The generalized total graph GTH(R) of R is the simple undirected graph with vertex set R and two distinct vertices x and y are adjacent if and only if x + y ∈ H. If we take R as the field F and H = {0}, we designate the graph as the generalized total graph of the field F and denote the same as GT (F ). In this paper, we investigate several graph theoretical properties of the generalized total graph GT (F ) and its complement GT (F ). In particular, we discuss about properties like Eulerian and Hamiltonian for GT (F ).</p><div class="ds-related-work--ctas"><button class="ds2-5-text-link ds2-5-text-link--inline js-swp-download-button" data-signup-modal="{"location":"wsj-grid-card-download-pdf-modal","work_title":"On the Generalized Total Graph of Fields and Its Complement","attachmentId":77164227,"attachmentType":"pdf","work_url":"https://www.academia.edu/65667420/On_the_Generalized_Total_Graph_of_Fields_and_Its_Complement","alternativeTracking":true}"><span class="material-symbols-outlined" style="font-size: 18px" translate="no">download</span><span class="ds2-5-text-link__content">Download free PDF</span></button><a class="ds2-5-text-link ds2-5-text-link--inline js-wsj-grid-card-view-pdf" href="https://www.academia.edu/65667420/On_the_Generalized_Total_Graph_of_Fields_and_Its_Complement"><span class="ds2-5-text-link__content">View PDF</span><span class="material-symbols-outlined" style="font-size: 18px" translate="no">chevron_right</span></a></div></div></div></div><div class="ds-sticky-ctas--wrapper js-loswp-sticky-ctas hidden"><div class="ds-sticky-ctas--grid-container"><div class="ds-sticky-ctas--container"><button class="ds2-5-button js-swp-download-button" data-signup-modal="{"location":"continue-reading-button--sticky-ctas","attachmentId":119841121,"attachmentType":"pdf","workUrl":null}">See full PDF</button><button class="ds2-5-button ds2-5-button--secondary js-swp-download-button" data-signup-modal="{"location":"download-pdf-button--sticky-ctas","attachmentId":119841121,"attachmentType":"pdf","workUrl":null}"><span class="material-symbols-outlined" style="font-size: 20px" translate="no">download</span>Download PDF</button></div></div></div><div class="ds-below-fold--grid-container"><div class="ds-work--container js-loswp-embedded-document"><div class="attachment_preview" data-attachment="Attachment_119841121" style="display: none"><div class="js-scribd-document-container"><div class="scribd--document-loading js-scribd-document-loader" style="display: block;"><img alt="Loading..." src="//a.academia-assets.com/images/loaders/paper-load.gif" /><p>Loading Preview</p></div></div><div style="text-align: center;"><div class="scribd--no-preview-alert js-preview-unavailable"><p>Sorry, preview is currently unavailable. You can download the paper by clicking the button above.</p></div></div></div></div><div class="ds-sidebar--container js-work-sidebar"><div class="ds-related-content--container"><h2 class="ds-related-content--heading">Related papers</h2><div class="ds-related-work--container js-related-work-sidebar-card" data-collection-position="0" data-entity-id="61008055" data-sort-order="default"><a class="ds-related-work--title js-related-work-grid-card-title ds2-5-body-md ds2-5-body-link" href="https://www.academia.edu/61008055/Structural_properties_of_the_graph_algebra">Structural properties of the graph algebra</a><div class="ds-related-work--metadata"><a class="js-related-work-grid-card-author ds2-5-body-sm ds2-5-body-link" data-author-id="72062292" href="https://independent.academia.edu/DavidNacin">David Nacin</a></div><p class="ds-related-work--metadata ds2-5-body-xs">Journal of Pure and Applied Algebra, 2008</p><div class="ds-related-work--ctas"><button class="ds2-5-text-link 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