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

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.