Some Families of Orthogonal Polynomials of a Discrete Variable and their Applications to Graphs and Codes

We present some related families of orthogonal polynomials of a discrete variable and survey some of their applications in the study of (distance-regular) graphs and (completely regular) codes. One of the main peculiarities of such orthogonal systems is their non-standard normalization condition, requiring that the square norm of each polynomial must equal its value at a given point of the mesh. For instance, when they are defined from the spectrum of a graph, one of these families is the system of the predistance polinomials which, in the case of distance-regular graphs, turns out to be the sequence of distance polinomials. The applications range from (quasi-spectral) characterizations of distance-regular graphs, walk-regular graphs, local distance-regularity and completely regular codes, to some results on representation theory.