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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 \(i=1,3\).

05E30 Association schemes, strongly regular graphs
Full Text: DOI
[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
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