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On the multiplicity of eigenvalues of distance-regular graphs

โœ Scribed by C.D. Godsil; J.H. Koolen


Publisher
Elsevier Science
Year
1995
Tongue
English
Weight
178 KB
Volume
226-228
Category
Article
ISSN
0024-3795

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โœ R.R. Zhu ๐Ÿ“‚ Article ๐Ÿ“… 1993 ๐Ÿ› Elsevier Science ๐ŸŒ English โš– 939 KB

Let \(G\) be a distance-regular graph. If \(G\) has an eigenvalue \(\theta\) of multiplicity \(m\) \((\geqslant 2)\), then \(G\) has a natural representation in \(R^{m}\). By studying the geometric properties of the image configuration in \(R^{m}\), we can obtain considerable information about the g

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We show that, if a bipartite distance-regular graph of valency k has an eigenvalue of multiplicity k, then it becomes 2-homogeneous. Combined with a result on bipartite 2-homogeneous distance-regular graphs by K. Nomura, we have a classification of such graphs.

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and Terwilliger recently introduced the notion of a tridiagonal pair. We apply their results to distance-regular graphs and obtain the following theorem. THEOREM. Let denote a distance-regular graph with diameter D โ‰ฅ 3. Suppose is Q-polynomial with respect to the ordering E 0 , E 1 , . . . , E D of

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In [1] N.L. Biggs mentions two parameter sets for distance regular graphs that are antipodal covers of a complete graph, for which existence of a corresponding graph was unknown. Here we settle both cases by proving that one does not exist, while there are exactly two nonisomorphic solutions to the

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Let denote a bipartite distance-regular graph with diameter D โ‰ฅ 4 and valency k โ‰ฅ 3. Let ฮธ 0 > ฮธ 1 > โ€ข โ€ข โ€ข > ฮธ D denote the eigenvalues of and let E 0 , E 1 , . . . , E D denote the associated primitive idempotents. Fix s (1 โ‰ค s โ‰ค D -1) and abbreviate E := E s . We say E is a tail whenever the entry