Positivity of Gibbs states on distance-regular graphs

We study criteria which ensure that Gibbs states (often also called generalized vacuum states) on distance-regular graphs are positive. Our main criterion assumes the graph can be embedded into a growing family of graphs. For proof positivity we then use polynomial hypergroup theory and translate this problem whether for [Formula: see text] function has positive integral representation w.r.t. orthogonal polynomials associated with graph. apply our to several examples. Hamming infinite distance-transitive obtain complete description states.