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| Main Authors: | , , |
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| Format: | Preprint |
| Published: |
2026
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2603.22601 |
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| _version_ | 1866917359241920512 |
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| author | van Dam, Edwin R. Monzillo, Giusy Penjić, Safet |
| author_facet | van Dam, Edwin R. Monzillo, Giusy Penjić, Safet |
| contents | Let $Γ$ be a connected regular graph with an eigenvalue $λ$ and corresponding idempotent $E_λ$. Let ${\cal E}_λ=\langle J,E_λ\rangle^\circ$ be the algebra generated by $J$ and $E_λ$ with respect to the entrywise-Hadamard product, where $J$ is the all-$1$ matrix. We study the combinatorial structure of a graph $Γ$ for which ${\cal E}_λ$ has dimension $2$, giving a combinatorial characterization of such graphs in terms of equitable partitions. We present many examples and classify the distance-regular graphs with this property, as well as graphs that generate a $3$-class association scheme. We also study the graphs that have two eigenvalues $λ$ for which ${\rm dim}({\cal E}_λ)=2$ and determine all such graphs with four distinct eigenvalues. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2603_22601 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | On the combinatorial structure of graphs with a spectral idempotent of small dual diameter van Dam, Edwin R. Monzillo, Giusy Penjić, Safet Combinatorics Let $Γ$ be a connected regular graph with an eigenvalue $λ$ and corresponding idempotent $E_λ$. Let ${\cal E}_λ=\langle J,E_λ\rangle^\circ$ be the algebra generated by $J$ and $E_λ$ with respect to the entrywise-Hadamard product, where $J$ is the all-$1$ matrix. We study the combinatorial structure of a graph $Γ$ for which ${\cal E}_λ$ has dimension $2$, giving a combinatorial characterization of such graphs in terms of equitable partitions. We present many examples and classify the distance-regular graphs with this property, as well as graphs that generate a $3$-class association scheme. We also study the graphs that have two eigenvalues $λ$ for which ${\rm dim}({\cal E}_λ)=2$ and determine all such graphs with four distinct eigenvalues. |
| title | On the combinatorial structure of graphs with a spectral idempotent of small dual diameter |
| topic | Combinatorics |
| url | https://arxiv.org/abs/2603.22601 |