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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:
05E30 Association schemes, strongly regular graphs
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