×

zbMATH — the first resource for mathematics

An inequality on the cosines of a tight distance-regular graph. (English) Zbl 0979.05112
The author considers a distance-regular graph \(\Gamma\) with diameter \(d\geq 3\), valency \(k\), and eigenvalues \(k=\theta_0>\theta_1>\cdots>\theta_d\), which is tight in the sense of A. Jurišić, J. Koolen, and P. Terwilliger [J. Algebr. Comb. 12, No. 2, 163-197 (2000; Zbl 0959.05121)]. He obtains an inequality involving the first, second and third cosines associated with \(\theta\), when \(\theta\) is \(\theta_1\) or \(\theta_d.\) (\(\theta_1\) and \(\theta_d\) are involved in the definition of a tight graph.) Also, the author proves that equality is attained if and only if \(\Gamma\) is dual biparite \(\mathcal Q\)-polynomial with respect to \(\theta\).

MSC:
05E30 Association schemes, strongly regular graphs
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] E. Bannai, T. Ito, Algebraic Combinatorics I: Association Schemes, The Benjamin-Cummings Publishing, Menlo Park, CA, 1984 · Zbl 0555.05019
[2] Curtin, B., 2-homogeneous bipartite distance-regular graphs, Discrete math., 187, 39-70, (1998) · Zbl 0958.05143
[3] Dickie, G.A.; Terwilliger, P.M., Dual bipartite \(Q\)-polynomial distance-regular graphs, European J. combin., 17, 613-623, (1996) · Zbl 0921.05064
[4] A.E. Brouwer, A.M. Cohen, A. Neumaier, Distance-Regular Graphs, Springer, Berlin, 1989 · Zbl 0747.05073
[5] C.D. Godsil, Algebraic Combinatorics, Chapman and Hall, New York, 1993
[6] A. Juris̆ić, J. Koolen, A local approach to 1-homogeneous graphs, Designs, Codes and Crytography 21 (2000) 127-147
[7] A. Juris̆ić, J. Koolen, P.M. Terwilliger, Tight distance-regular graphs, J. Algebraic Combin. 12 (2000) 163-197 · Zbl 0959.05121
[8] A. Juris̆ić, J. Koolen, P.M. Terwilliger, Tight distance-regular graphs with small diameter, University of Ljubljana Preprint Series 36 (621) (1998)
[9] M. Maclean, An inequality involving two eigenvalues of a bipartite distance-regular graph, Discrete Math. 225 (2000) 193-216 · Zbl 1001.05124
[10] Pascasio, A.A., Tight graphs and their primitive idempotents, J. algebraic combin., 10, 1, 47-59, (1999) · Zbl 0927.05085
[11] A.A. Pascasio, Tight distance-regular graphs and the Q-polynomial property, Graphs Combin. (to appear) · Zbl 0993.05147
[12] Terwilliger, P.M., Balanced sets and Q-polynomial association schemes, Graphs combin., 4, 87-94, (1988) · Zbl 0644.05016
[13] Terwilliger, P.M., A new inequality for distance-regular graphs, Discrete math., 137, 319-332, (1995) · Zbl 0814.05074
[14] M. Tomiyama, A note on the primitive idempotents of distance-regular graphs (submitted) · Zbl 0993.05148
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.