From regular boundary graphs to antipodal distance-regular graphs

Let Γ be a regular graph with n vertices, diameter D, and d + 1 different eigenvalues λ > λ1 > · · · > λd. In a previous paper, the authors showed that if P (λ) > n − 1, then D ≤ d − 1, where P is the polynomial of degree d−1 which takes alternating values±1 atλ1, . . . , λd. The graphs satisfying P (λ) = n − 1, called boundary graphs, have shown to deserve some attention because of their rich structure. This paper is devoted to the study of this case and, as a main result, it is shown that those extremal (D = d) boundary graphs where each vertex have maximum eccentricity are, in fact, 2-antipodal distanceregular graphs. The study is carried out by using a new sequence of orthogonal polynomials, whose special properties are shown to be induced by their intrinsic symmetry. c © 1998 John Wiley & Sons, Inc. J Graph Theory 27: 123–140, 1998