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Nonexistence of some antipodal distance-regular graphs of diameter four. (English) Zbl 0958.05139
The authors show that for distance-regular graphs with certain intersection arrays, the first subconstituent graphs are strongly regular. Theorem 2.2. Let $$\Gamma$$ be a nonbipartite distance-regular graph with diameter $$d\geq 3$$, eigenvalues $$k=\theta_0>\cdots>\theta_d$$, and let $$b^-=-1-b_1/(\theta_1+1)$$, $$b^+=-1-b_1/(\theta_d+1)$$. Then $$k(a_1+b^+b^-)\leq (a_1-b^+)(a_1-b^-)$$, and equality holds if and only if all local graphs are connected strongly regular graphs with eigenvalues $$a_1,b^+,b^-$$.
Let $$\Gamma$$ be a distance-regular graph, whose local graphs are strongly regular with parameters $$(k',\lambda',\mu')$$. Then the $$\mu$$-graphs of $$\Gamma$$ are regular with valency $$\mu'$$, $$c_2\mu'$$ is even and $$c_2\geq \mu'+1$$, with equality if and only if $$\Gamma$$ is a Terwilliger graph (Theorem 3.1).
Corollary 3.5. Let $$\Gamma$$ be a nonbipartite antipodal distance-regular graph with diameter four and covering index $$r$$ and $$k(a_1+b^+ b^-)=(a_1-b^+)(a_1-b^-)$$. Then $$b^+$$ and $$b^-$$ are integral, $$b^+\geq 1$$, $$b^-\leq -2$$ and $$r$$ divides $$b^+-b^-$$.
Theorem 3.1 and Corollary 3.5 give new existence conditions for the corresponding distance-regular graphs. In particular 20 intersection arrays from tables of feasible parameters of nonbipartite antipodal distance-regular graphs with diameter 4 are ruled out.

##### MSC:
 5e+30 Association schemes, strongly regular graphs
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##### References:
 [1] Biggs, N.L., Algebraic graph theory, (1974), Cambridge University Press Cambridge · Zbl 0501.05039 [2] A. E. Brouwer [3] Brouwer, A.E.; Cohen, A.M.; Neumaier, A., Distance-regular graphs, (1989), Springer-Verlag Berlin, Heidelberg · Zbl 0747.05073 [4] A. E. Brouwer, A. M. Cohen, A. Neumaier [5] Dickie, G.A.; Terwilliger, P.M., Dual bipartite Q -polynomial distance-regular graphs, Europ. J. combinatorics, 17, 613-623, (1996) · Zbl 0921.05064 [6] Godsil, C.D., Algebraic combinatorics, (1993), Chapman and Hall New York · Zbl 0814.05075 [7] W. H. Haemers, 1979 [8] W. H. Haemers, Interlacing eigenvalues and graphs, Linear Algebr. Appl. 226, 228 · Zbl 0831.05044 [9] 1995, 593, 616 [10] Jurišić, A.; Koolen, J.; Terwilliger, P., Tight distance-regular graphs, J. algebr. comb., 12, 163-197, (2000) · Zbl 0959.05121 [11] Jurišić, A.; Koolen, J.; Terwilliger, P., Krein parameters and antipodal tight graphs with diameter 3 and 4, Discrete. math., submitted, (1999) [12] Meixner, T., Some polar towers, Europ. J. combinatorics, 12, 397-415, (1991) · Zbl 0753.05016 [13] Soicher, L.H., Three new distance-regular graphs, Europ. J. combinatorics, 14, 501-505, (1993) · Zbl 0794.05135
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