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 ( 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.