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EMIS Electronic Library of Mathematics (ELibM)
The Open Access Repository of Mathematics |
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JOURNAL OF
ALGEBRAIC
COMBINATORICS
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Editors-in-chief: C. A. Athanasiadis, T. Lam, A. Munemasa, H. Van Maldeghem
ISSN 0925-9899 (print) • ISSN 1572-9192 (electronic) |
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Taut Distance-Regular Graphs of Odd Diameter
Mark S. MacLean
University of North Carolina Asheville NC 28804 USA
DOI: 10.1023/A:1022926629298
AbstractLet _boxclose _i - _i - 1 = , = \begin{gathered} \frac{{σ_{i + 1} - ασ}}{{σσ_i - σ_{i - 1} }} = \frac{{βρ_i - ρ_{i - 1} }}{{ρρ_i - ρ_{i - 1} }}, \hfill \\ \frac{{ρ_{i + 1} - βρ_i }}{{ρρ_i - ρ_{i - 1} }} = \frac{{ασ_i - σ_{i - 1} }}{{σσ_i - σ_{i - 1} }} \hfill \\ \end{gathered} |
for 1 
i D - 1, where 
= 1, 
= 1. Using these equations, we recursively obtain 0, 1, ..., D and 0, 1, ..., D in terms of the four real scalars , , 
, 
. From this we obtain all intersection numbers of 
in terms of , , , . We showed in an earlier paper that the pair E 1, E d is taut, where d = ( D - 1)/2. Applying our results to this pair, we obtain the intersection numbers of in terms of k, 
, 
1, d, where denotes the intersection number c 2. We show that if is taut and D is odd, then is an antipodal 2-cover.
Pages: 125–147
Keywords: distance-regular graph; association scheme; bipartite graph; tight graph; taut graph
Full Text: PDF
References
1. E. Bannai and T. Ito, Algebraic Combinatorics I: Association Schemes, Benjamin/Cummings, London, 1984.
2. A.E. Brouwer, A.M. Cohen, and A. Neumaier, Distance-Regular Graphs, Springer-Verlag, Berlin, 1989.
3. B. Curtin, “2-homogeneous bipartite distance-regular graphs,” Discrete Math. 187 (1998), 39-70.
4. G. Dickie and P. Terwilliger, “Dual bipartite Q-polynomial distance-regular graphs,” Europ. J. Combin. 17 (1996), 613-623.
5. C.D. Godsil, Algebraic Combinatorics, Chapman and Hall, New York, 1993.
6. A. Juri\check sić and J. Koolen, “1-homogeneous graphs with Cocktail Party μ-graphs,” J. Alg. Combin., in press.
7. A. Juri\check sić and J. Koolen, “Krein parameters and antipodal tight graphs with diameter 3 and 4,” Discrete Math. 244 (2002), 181-202.
8. A. Juri\check sić and J. Koolen, “Nonexistence of some antipodal distance-regular graphs of diameter four,” Europ. J. Combin. 21 (2000), 1039-1046.
9. A. Juri\check sić and J. Koolen, “A local approach to 1-homogeneous graphs,” Designs, Codes, and Cryptography 21 (2000), 127-147.
10. A. Juri\check sić, J. Koolen, and P. Terwilliger, “Tight distance-regular graphs with small diameter,” University of Ljubljana Preprint Series 36 (621), 1998.
11. A. Juri\check sić, J. Koolen, and P. Terwilliger, “Tight distance-regular graphs,” J. Alg. Combin. 12 (2000), 163-197.
12. M. MacLean, “An inequality involving two eigenvalues of a bipartite distance-regular graph,” Discrete Math. 225 (2000), 193-216.
13. K. Nomura, “Homogeneous graphs and regular near polygons,” J. Combin. Theory Ser. B 60 (1994), 63-71.
14. K. Nomura, “Spin models on bipartite distance-regular graphs,” J. Combin. Theory Ser. B 64 (1995), 300-313.
15. A. Pascasio, “An inequality in character algebras,” Discrete Math., in press.
16. A. Pascasio, “An inequality on the cosines of a tight distance-regular graph,” Linear Algebra Appl. 325 (2001), 147-159.
17. A. Pascasio, “Tight distance-regular graphs and the Q-polynomial property,” Graphs Combin. 17 (2001), 149-169.
18. A. Pascasio, “Tight graphs and their primitive idempotents,” J. Alg. Combin. 10 (1999), 47-59.
19. M. Tomiyama, “A note on the primitive idempotents of distance-regular graphs,” Discrete Math. 240 (2001), 281-294.
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