Conformally rigid graphs
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866908301844807680 |
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| author | Steinerberger, Stefan Thomas, Rekha R. |
| author_facet | Steinerberger, Stefan Thomas, Rekha R. |
| contents | Given a finite, simple, connected graph $G=(V,E)$ with $|V|=n$, we consider the associated graph Laplacian matrix $L = D - A$ with eigenvalues $0 = λ_1 < λ_2 \leq \dots \leq λ_n$. One can also consider the same graph equipped with positive edge weights $w:E \rightarrow \mathbb{R}_{> 0}$ normalized to $\sum_{e \in E} w_e = |E|$ and the associated weighted Laplacian matrix $L_w$. We say that $G$ is conformally rigid if constant edge-weights maximize the second eigenvalue $λ_2(w)$ of $L_w$ over all $w$, and minimize $λ_n(w')$ of $L_{w'}$ over all $w'$, i.e., for all $w,w'$, $$ λ_2(w) \leq λ_2(1) \leq λ_n(1) \leq λ_n(w').$$ Conformal rigidity requires an extraordinary amount of symmetry in $G$. Every edge-transitive graph is conformally rigid. We prove that every distance-regular graph, and hence every strongly-regular graph, is conformally rigid. Certain special graph embeddings can be used to characterize conformal rigidity. Cayley graphs can be conformally rigid but need not be, we prove a sufficient criterion. We also find a small set of conformally rigid graphs that do not belong into any of the above categories; these include the Hoffman graph, the crossing number graph 6B and others. Conformal rigidity can be certified via semidefinite programming, we provide explicit examples. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2402_11758 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Conformally rigid graphs Steinerberger, Stefan Thomas, Rekha R. Combinatorics Discrete Mathematics Optimization and Control Spectral Theory Given a finite, simple, connected graph $G=(V,E)$ with $|V|=n$, we consider the associated graph Laplacian matrix $L = D - A$ with eigenvalues $0 = λ_1 < λ_2 \leq \dots \leq λ_n$. One can also consider the same graph equipped with positive edge weights $w:E \rightarrow \mathbb{R}_{> 0}$ normalized to $\sum_{e \in E} w_e = |E|$ and the associated weighted Laplacian matrix $L_w$. We say that $G$ is conformally rigid if constant edge-weights maximize the second eigenvalue $λ_2(w)$ of $L_w$ over all $w$, and minimize $λ_n(w')$ of $L_{w'}$ over all $w'$, i.e., for all $w,w'$, $$ λ_2(w) \leq λ_2(1) \leq λ_n(1) \leq λ_n(w').$$ Conformal rigidity requires an extraordinary amount of symmetry in $G$. Every edge-transitive graph is conformally rigid. We prove that every distance-regular graph, and hence every strongly-regular graph, is conformally rigid. Certain special graph embeddings can be used to characterize conformal rigidity. Cayley graphs can be conformally rigid but need not be, we prove a sufficient criterion. We also find a small set of conformally rigid graphs that do not belong into any of the above categories; these include the Hoffman graph, the crossing number graph 6B and others. Conformal rigidity can be certified via semidefinite programming, we provide explicit examples. |
| title | Conformally rigid graphs |
| topic | Combinatorics Discrete Mathematics Optimization and Control Spectral Theory |
| url | https://arxiv.org/abs/2402.11758 |