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An association scheme is $P$-polynomial if and only if it consists of thedistance matrices of a distance-regular graph. Recently, multivariate$P$-polynomial association schemes have been introduced. In this paper, thenotion of $m$-distance-regular graph is defined and shown to give a graphinterpretation of the multivariate $P$-polynomial association schemes. Variousexamples are provided, along with a discussion of bivariate $P$-polynomialschemes of type $(\alpha, \beta)$ and their connection to $(\alpha,\beta)$-compatible 2-distance-regular graphs.

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