Conformal Rigidity of Graphs: Subdifferentials and Orbit-Isometries
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arXiv
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| Formato: | Preprint |
| Publicado: |
2026
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| _version_ | 1866910222111473664 |
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| author | Niu, Andrew |
| author_facet | Niu, Andrew |
| contents | A connected undirected graph $G = (V,E)$ is lower conformally rigid if uniform edge weights maximize the second smallest Laplacian eigenvalue $λ_2(w)$ over all normalized edge weights $w$, and upper conformally rigid if uniform edge weights minimize the largest eigenvalue $λ_n(w)$ 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 spectral embeddings, which are known to characterize conformal rigidity. This subdifferential framework lends itself naturally to techniques of symmetry reduction that motivate the notion of an orbit-isometric embedding - a weaker condition than edge-isometry that accounts for the symmetries of $G$ while remaining sufficient for conformal rigidity. The notion opens the door to tools from representation theory: 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. This extra structure enables a new, algebraically exact certification method for conformal rigidity, bypassing the numerical difficulties of prior approaches. In many cases, the problem reduces to a check of linear feasibility, and in general, to solving a system of quadratic equations via Gröbner bases. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2605_15017 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Conformal Rigidity of Graphs: Subdifferentials and Orbit-Isometries Niu, Andrew Combinatorics Optimization and Control Spectral Theory A connected undirected graph $G = (V,E)$ is lower conformally rigid if uniform edge weights maximize the second smallest Laplacian eigenvalue $λ_2(w)$ over all normalized edge weights $w$, and upper conformally rigid if uniform edge weights minimize the largest eigenvalue $λ_n(w)$ 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 spectral embeddings, which are known to characterize conformal rigidity. This subdifferential framework lends itself naturally to techniques of symmetry reduction that motivate the notion of an orbit-isometric embedding - a weaker condition than edge-isometry that accounts for the symmetries of $G$ while remaining sufficient for conformal rigidity. The notion opens the door to tools from representation theory: 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. This extra structure enables a new, algebraically exact certification method for conformal rigidity, bypassing the numerical difficulties of prior approaches. In many cases, the problem reduces to a check of linear feasibility, and in general, to solving a system of quadratic equations via Gröbner bases. |
| title | Conformal Rigidity of Graphs: Subdifferentials and Orbit-Isometries |
| topic | Combinatorics Optimization and Control Spectral Theory |
| url | https://arxiv.org/abs/2605.15017 |