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

Abstract

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 polynomials which, in the case of distance-regular graphs, turns out to be the sequence of distance polynomials. 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.

On the k-independence number of graphs

“…and satisfy 1 = q 0 (θ 0 ) < q 1 (θ 0 ) < · · · < q d−1 (θ 0 ) < n. See [5] for further details and applications.…”

Some spectral and quasi-spectral characterizations of distance-regular graphs

“…For convenience, we also define the preintersection numbers γ 0 = 0 and β d = 0. Some basic properties of the predistance polynomials and preintersection numbers are included in the following result (see Cámara, Fàbrega, Fiol, and Garriga [5]).…”

The spectral excess theorem for distance-regular graphs having distance-d graph with fewer distinct eigenvalues

“…. Some interesting consequences of the above, together with other properties of the predistance polynomials are the following (for more details, see [4]):…”