×

zbMATH — the first resource for mathematics

Taut distance-regular graphs of even diameter. (English) Zbl 1047.05044
Let \(\Gamma\) be a distance-regular graph with diameter \(D\geq 4\) and valency \(k\geq 3\). Let \(k=\theta_0>\cdots >\theta_D\) denote the distinct eigenvalues of \(\Gamma\), and for \(0\leq i\leq D\), let \(E_i\) denote the primitive idempotents of \(\Gamma\) associated with \(\theta_i\). A primitive idempotent \(F\) of \(\Gamma\) is called trivial, if \(F=E_0\) or \(\Gamma\) is bipartite and \(F=E_D\). Let \(E,F\) denote primitive idempotents of \(\Gamma\). The pair \(E,F\) is called taut whenever \(E,F\) are nontrivial and the entry-wise product \(E\circ F\) is a linear combination of two distinct primitive idempotents of \(\Gamma\). We define \(\Gamma\) to be taut, if \(\Gamma\) is not 2-homogeneous and \(\Gamma\) has the taut pair \(E,F\) of primitive idempotents. Theorem. Let \(\Gamma\) be a bipartite distance-regular graph with even diameter \(D\geq 4\), valency \(k\geq 3\), and eigenvalues \(\theta_0>\dots >\theta_D\). Let \(\theta\) denote an eigenvalue of \(\Gamma\) other than \(\theta_0,\theta_D\), and let \(\sigma_0,\dots ,\sigma_D\) denote the cosine sequence associated with \(\theta\). Then the following are equivalent: (i) \(\Gamma\) is taut or 2-homogeneous, and \(\theta\in \{\theta_1,\theta_{D-1}\}\). (ii) There exists a complex number \(\lambda\) such that \(\sigma_{i-1}-\lambda \sigma_i+\sigma_{i+1}=0\) for \(i\) odd, \(1\leq i\leq D-1\). (iii) There exists a complex number \(\lambda\) such that \(\sigma_{i-1}-\lambda \sigma_i+\sigma_{i+1}=0\) for \(i=1,3\).

MSC:
05E30 Association schemes, strongly regular graphs
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] Bannai, E.; Ito, T., Algebraic combinatorics I: association schemes, (1984), Benjamin/Cummings London · Zbl 0555.05019
[2] Brouwer, A.E.; Cohen, A.M.; Neumaier, A., Distance-regular graphs, (1989), Springer Berlin · Zbl 0747.05073
[3] Curtin, B., 2-homogeneous bipartite distance-regular graphs, Discrete math., 187, 39-70, (1998) · Zbl 0958.05143
[4] Dickie, G.; Terwilliger, P., Dual bipartite Q-polynomial distance-regular graphs, European J. combin., 17, 613-623, (1996) · Zbl 0921.05064
[5] Godsil, C.D., Algebraic combinatorics, (1993), Chapman & Hall New York · Zbl 0814.05075
[6] Hemmeter, J., Distance-regular graphs and halved graphs, European J. combin., 7, 119-129, (1986) · Zbl 0606.05041
[7] Jurišić, A.; Koolen, J., 1-homogeneous graphs with cocktail party μ-graphs, J. algebraic combin., 18, 79-98, (2003) · Zbl 1038.05059
[8] Jurišić, A.; Koolen, J., A local approach to 1-homogeneous graphs, Des. codes cryptogr., 21, 127-147, (2000) · Zbl 0964.05073
[9] Jurišić, A.; Koolen, J., Nonexistence of some antipodal distance-regular graphs of diameter four, European J. combin., 21, 1039-1046, (2000) · Zbl 0958.05139
[10] Jurišić, A.; Koolen, J., Krein parameters and antipodal tight graphs with diameter 3 and 4, Discrete math., 244, 181-202, (2002) · Zbl 1024.05086
[11] Jurišić, A.; Koolen, J.; Terwilliger, P., Tight distance-regular graphs with small diameter, Univ. Ljubljana preprint ser., 36, 621, 1-21, (1998)
[12] Jurišić, A.; Koolen, J.; Terwilliger, P., Tight distance-regular graphs, J. algebraic combin., 12, 163-197, (2000) · Zbl 0959.05121
[13] MacLean, M., An inequality involving two eigenvalues of a bipartite distance-regular graph, Discrete math., 225, 193-216, (2000) · Zbl 1001.05124
[14] MacLean, M., Taut distance-regular graphs of odd diameter, J. algebraic combin., 17, 125-147, (2003) · Zbl 1014.05072
[15] Nomura, K., Homogeneous graphs and regular near polygons, J. combin. theory ser. B, 60, 63-71, (1994) · Zbl 0793.05130
[16] Nomura, K., Spin models on bipartite distance-regular graphs, J. combin. theory ser. B, 64, 300-313, (1995) · Zbl 0827.05060
[17] Pascasio, A., An inequality in character algebras, Discrete math., 264, 201-210, (2003) · Zbl 1014.05076
[18] Pascasio, A., Tight distance-regular graphs and the Q-polynomial property, Graphs combin., 17, 149-169, (2001) · Zbl 0993.05147
[19] Pascasio, A., Tight graphs and their primitive idempotents, J. algebraic combin., 10, 47-59, (1999) · Zbl 0927.05085
[20] Pascasio, A., An inequality on the cosines of a tight distance-regular graph, Linear algebra appl., 325, 147-159, (2001) · Zbl 0979.05112
[21] Terwilliger, P., The subconstituent algebra of an association scheme I, J. algebraic combin., 1, 363-388, (1992) · Zbl 0785.05089
[22] Tomiyama, M., On the primitive idempotents of distance-regular graphs, Discrete math., 240, 281-294, (2001) · 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.