  We study criteria which ensure that Gibbs states (often also called
generalized vacuum states) on distance-regular graphs are positive. Our main
criterion assumes that the graph can be embedded into a growing family of
distance-regular graphs. For the proof of the positivity we then use polynomial
hypergroup theory and translate this positivity into the problem whether for
$x\in[-1,1]$ the function $n\mapsto x^n$ has a positive integral representation
w.r.t. the orthogonal polynomials associated with the graph. We apply our
criteria to several examples. For Hamming graphs and the infinite
distance-transitive graphs we obtain a complete description of the positive
Gibbs states.
