Distance-regular subgraphs in a distance-regular graph. II (Q1902966)

Let \(\Gamma\) be a distance-regular graph with intersection array \(\{b_0, \dots, b_{d - 1}; c_1, \dots, c_d\}\). Then \(\Gamma\) is regular with \(k = b_0\). Set \(a_j = k_j - b_j - c_j\), \(l(\alpha, \beta, \gamma) = |\{j \mid (c_j, a_j, b_j) = (\alpha, \beta, \gamma)\} |\) . It is known that a distance-regular graph with \(r = l(1, a_1, b_1) = 1\) and \(c_3 = 1\) contains a distance- regular subgraph of diameter 2 with \(c_2 = 1\). The author generalizes this result to general \(r\) (the case \(a_1 = 0\) was considered in part I, see Zbl 0836.05080 above). Theorem. Let \(\Gamma\) be a distance-regular graph with \(r = l(1, a_1, b_1)\). If \(a_1 > 0\) and \(c_{2r + 1} = 1\) then \(\Gamma\) contains a Moore subgraph of valency \(a_{r + 1} + 1\) and diameter \(r + 1\). As a corollary if \(\Gamma\) be a distance-regular graph with \(a_1 > 0\), \(r = l(1, a_1, b_1) \geq 2\), and \(c_s = 1\), then \(s \leq 2r\).