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arXiv · 2309.16016

m-distance-regular graphs and their relation to multivariate P-polynomial association schemes

Pierre-Antoine Bernard·Nicolas Crampe·Luc Vinet·Meri Zaimi·Xiaohong Zhang
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Abstract

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 $(\alpha,\beta)$ 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 $(\alpha, \beta)$ are discussed, and their connection to 2-distance-regular graphs is established.

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Pierre-Antoine Bernard, Nicolas Crampe, Luc Vinet, Meri Zaimi, Xiaohong Zhang. 2023-09-27. m-distance-regular graphs and their relation to multivariate P-polynomial association schemes. https://arxiv.org/abs/2309.16016

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