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| Main Authors: | , , , , |
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| Format: | Preprint |
| Published: |
2023
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2309.16016 |
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| _version_ | 1866909085695213568 |
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| author | Bernard, Pierre-Antoine Crampe, Nicolas Vinet, Luc Zaimi, Meri Zhang, Xiaohong |
| author_facet | Bernard, Pierre-Antoine Crampe, Nicolas Vinet, Luc Zaimi, Meri Zhang, Xiaohong |
| contents | An association scheme is $P$-polynomial if and only if it consists of the distance matrices of a distance-regular graph. Recently, bivariate $P$-polynomial association schemes of type $(α,β)$ were introduced by Bernard et al., and multivariate $P$-polynomial association schemes were later defined by Bannai et al. In this paper, the notion of $m$-distance-regular graph is defined and shown to give a graph interpretation of the multivariate $P$-polynomial association schemes. Various examples are provided. Refined structures and additional constraints for multivariate $P$-polynomial association schemes and $m$-distance-regular graphs are also considered. In particular, bivariate $P$-polynomial schemes of type $(α, β)$ are discussed, and their connection to 2-distance-regular graphs is established. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2309_16016 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | m-distance-regular graphs and their relation to multivariate P-polynomial association schemes Bernard, Pierre-Antoine Crampe, Nicolas Vinet, Luc Zaimi, Meri Zhang, Xiaohong Combinatorics An association scheme is $P$-polynomial if and only if it consists of the distance matrices of a distance-regular graph. Recently, bivariate $P$-polynomial association schemes of type $(α,β)$ were introduced by Bernard et al., and multivariate $P$-polynomial association schemes were later defined by Bannai et al. In this paper, the notion of $m$-distance-regular graph is defined and shown to give a graph interpretation of the multivariate $P$-polynomial association schemes. Various examples are provided. Refined structures and additional constraints for multivariate $P$-polynomial association schemes and $m$-distance-regular graphs are also considered. In particular, bivariate $P$-polynomial schemes of type $(α, β)$ are discussed, and their connection to 2-distance-regular graphs is established. |
| title | m-distance-regular graphs and their relation to multivariate P-polynomial association schemes |
| topic | Combinatorics |
| url | https://arxiv.org/abs/2309.16016 |