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From regular boundary graphs to antipodal distance-regular graphs

✍ Scribed by Fiol, M. A.; Garriga, E.; Yebra, J. L. A.

Publisher: John Wiley and Sons
Year: 1998
Tongue: English
Weight: 383 KB
Volume: 27
Category: Article
ISSN: 0364-9024
DOI: 10.1002/(sici)1097-0118(199803)27:3<123::aid-jgt2>3.0.co;2-q

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✦ Synopsis

Let Γ be a regular graph with n vertices, diameter D, and d + 1

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 at λ 1 , . . . , λ d . 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.

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