an:01545625 Zbl 0958.05139 Juri??i??, Aleksandar; Koolen, Jack Nonexistence of some antipodal distance-regular graphs of diameter four EN Eur. J. Comb. 21, No. 8, 1039-1046 (2000). 00070777 2000
j
05E30 distance-regular graphs; strongly regular graph; eigenvalues 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. Alexandre A.Makhnev (Ekaterinburg)