The Quad-$C_5$ Graph: Maximum Contextuality Gap on Eight Vertices
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Quantum Physics arXiv:2605.12828 (quant-ph) [Submitted on 12 May 2026] Title:The Quad-$C_5$ Graph: Maximum Contextuality Gap on Eight Vertices Authors:Ugur Tamer, Özgür E. Müstecaplıoğlu View a PDF of the paper titled The Quad-$C_5$ Graph: Maximum Contextuality Gap on Eight Vertices, by Ugur Tamer and 1 other authors View PDF HTML (experimental) Abstract:We perform an exhaustive semidefinite-programming search over all 11{,}117 connected non-isomorphic simple graphs on eight vertices to maximize the quantum contextuality gap $\Delta(G)=\vartheta(G)-\alpha(G)$, where $\vartheta(G)$ is the Lovász theta function and $\alpha(G)$ is the independence number of the exclusion graph $G$ within the Cabello--Severini--Winter framework for projective measurements. A previously uncharacterized graph on $n=8$ vertices and $m=10$ edges, which we name the Quad-$C_5$ graph (graph6 code: \texttt{GCQb`o}), achieves $\Delta=0.46784$, surpassing the Wagner graph $W$ ($\Delta\approx0.414$, $m=12$) with two fewer edges. We determine numerically, via the PSLQ integer-relation algorithm at 50-digit precision, that Quad-$C_5$ is a \emph{qutrit} contextuality witness with $\eta_3=1+\sqrt{5}$ (minimal polynomial $x^2-2x-4=0$), while numerical evidence indicates the Wagner graph requires a four-dimensional (two-qubit) Hilbert space. The graph contains four mutually overlapping induced five-cycles, and its adjacency spectrum is dominated by golden-ratio eigenvalues, tracing the contextuality advantage algebraically to the KCBS pentagon. Under depolarizing noise, Quad-$C_5$ at $d=3$ shares the critical visibility $v^*=1/(3\sqrt{5}-5)\approx0.585$ of the KCBS witness -- an analytically provable coincidence arising from a uniform shift of the graph parameters -- while at $d=4$ it strictly surpasses the Wagner graph in noise robustness. Comments: Subjects: Quantum Physics (quant-ph) Cite as: arXiv:2605.12828 [quant-ph] (or arXiv:2605.12828v1 [quant-ph] for this version) https://doi.org/10.48550/arXiv.2605.12828 Focus to learn more arXiv-issued DOI via DataCite (pending registration) Submission history From: Uğur Tamer [view email] [v1] Tue, 12 May 2026 23:50:25 UTC (16 KB) Full-text links: Access Paper: View a PDF of the paper titled The Quad-$C_5$ Graph: Maximum Contextuality Gap on Eight Vertices, by Ugur Tamer and 1 other authorsView PDFHTML (experimental)TeX Source view license Current browse context: quant-ph new | recent | 2026-05 References & Citations INSPIRE HEP NASA ADSGoogle Scholar Semantic Scholar export BibTeX citation Loading... BibTeX formatted citation × loading... Data provided by: Bookmark Bibliographic Tools Bibliographic and Citation Tools Bibliographic Explorer Toggle Bibliographic Explorer (What is the Explorer?) Connected Papers Toggle Connected Papers (What is Connected Papers?) Litmaps Toggle Litmaps (What is Litmaps?) scite.ai Toggle scite Smart Citations (What are Smart Citations?) Code, Data, Media Code, Data and Media Associated with this Article alphaXiv Toggle alphaXiv (What is alphaXiv?) Links to Code Toggle CatalyzeX Code Finder for Papers (What is CatalyzeX?) DagsHub Toggle DagsHub (What is DagsHub?) GotitPub Toggle Gotit.pub (What is GotitPub?) Huggingface Toggle Hugging Face (What is Huggingface?) ScienceCast Toggle ScienceCast (What is ScienceCast?) Demos Demos Replicate Toggle Replicate (What is Replicate?) Spaces Toggle Hugging Face Spaces (What is Spaces?) Spaces Toggle TXYZ.AI (What is TXYZ.AI?) Related Papers Recommenders and Search Tools Link to Influence Flower Influence Flower (What are Influence Flowers?) Core recommender toggle CORE Recommender (What is CORE?) Author Venue Institution Topic About arXivLabs arXivLabs: experimental projects with community collaborators arXivLabs is a framework that allows collaborators to develop and share new arXiv features directly on our website. Both individuals and organizations that work with arXivLabs have embraced and accepted our values of openness, community, excellence, and user data privacy. arXiv is committed to these values and only works with partners that adhere to them. Have an idea for a project that will add value for arXiv's community? Learn more about arXivLabs. Which authors of this paper are endorsers? | Disable MathJax (What is MathJax?)