{"paper":{"title":"Conformal Rigidity of Graphs: Subdifferentials and Orbit-Isometries","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"A single eigenvector certifies conformal rigidity for vertex-transitive graphs and similar symmetric ones.","cross_cats":["math.OC","math.SP"],"primary_cat":"math.CO","authors_text":"Andrew Niu","submitted_at":"2026-05-14T16:18:40Z","abstract_excerpt":"A connected undirected graph $G = (V,E)$ is lower conformally rigid if uniform edge weights maximize the second smallest Laplacian eigenvalue $\\lambda_2(w)$ over all normalized edge weights $w$, and upper conformally rigid if uniform edge weights minimize the largest eigenvalue $\\lambda_n(w)$ over all normalized edge weights; $G$ is conformally rigid if it is lower or upper conformally rigid. This paper establishes a new framework for conformal rigidity through the language of subdifferentials, unifying the variational perspective on eigenvalue optimization with the geometry of edge-isometric "},"claims":{"count":4,"items":[{"kind":"strongest_claim","text":"for a large class of graphs, including all vertex-transitive ones, we show that conformal rigidity is certified by a single eigenvector, resolving an open question and explaining the conformal rigidity of previously unexplained graphs.","source":"verdict.strongest_claim","status":"machine_extracted","claim_id":"C1","attestation":"unclaimed"},{"kind":"weakest_assumption","text":"The assumption that edge-isometric spectral embeddings characterize conformal rigidity (unified here with the subdifferential perspective) and that the orbit-isometric weakening remains sufficient.","source":"verdict.weakest_assumption","status":"machine_extracted","claim_id":"C2","attestation":"unclaimed"},{"kind":"one_line_summary","text":"A subdifferential framework certifies conformal rigidity via orbit-isometric embeddings, reducing the problem for vertex-transitive graphs to a single-eigenvector check and in general to linear feasibility or Gröbner bases.","source":"verdict.one_line_summary","status":"machine_extracted","claim_id":"C3","attestation":"unclaimed"},{"kind":"headline","text":"A single eigenvector certifies conformal rigidity for vertex-transitive graphs and similar symmetric ones.","source":"verdict.pith_extraction.headline","status":"machine_extracted","claim_id":"C4","attestation":"unclaimed"}],"snapshot_sha256":"5f5ea48f2e2e73db964e35c897d9c96f565f3ec024f8347e04dc8e2486e938bb"},"source":{"id":"2605.15017","kind":"arxiv","version":1},"verdict":{"id":"99a492e2-18d2-483b-8991-6a133a747be7","model_set":{"reader":"grok-4.3"},"created_at":"2026-05-15T03:04:25.095617Z","strongest_claim":"for a large class of graphs, including all vertex-transitive ones, we show that conformal rigidity is certified by a single eigenvector, resolving an open question and explaining the conformal rigidity of previously unexplained graphs.","one_line_summary":"A subdifferential framework certifies conformal rigidity via orbit-isometric embeddings, reducing the problem for vertex-transitive graphs to a single-eigenvector check and in general to linear feasibility or Gröbner bases.","pipeline_version":"pith-pipeline@v0.9.0","weakest_assumption":"The assumption that edge-isometric spectral embeddings characterize conformal rigidity (unified here with the subdifferential perspective) and that the orbit-isometric weakening remains sufficient.","pith_extraction_headline":"A single eigenvector certifies conformal rigidity for vertex-transitive graphs and similar symmetric ones."},"references":{"count":48,"sample":[{"doi":"","year":2025,"title":"Catherine Babecki, Stefan Steinerberger, and Rekha R. Thomas. Spectrahedral geometry of graph sparsifiers.SIAM J. Discrete Math., 39(1):449–483, 2025","work_id":"eac39494-d20b-4d41-bad6-8d3e790ee497","ref_index":1,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"","year":2019,"title":"Laugesen.Symmetrization in analysis","work_id":"0f726880-0e09-482c-8863-a258b4215604","ref_index":2,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"","year":2001,"title":"MPS/SIAM Series on Optimization","work_id":"a58e1014-1aed-4144-b235-f7285425d0b6","ref_index":3,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"","year":2024,"title":"A survey of maximalk-degenerate graphs andk-trees.Theory and Applications of Graphs, 0(1):Article 5, 2024","work_id":"427292b2-4312-4887-bc3b-939f0b49aa3a","ref_index":4,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"","year":1998,"title":"Jacek Bochnak, Michel Coste, and Marie-Fran¸ coise Roy.Real algebraic geometry. Transl. from the French., volume 36 ofErgeb. Math. Grenzgeb., 3. Folge. 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