Distance-regular subgraphs in a distance-regular graph. I (Q1902965)

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 \(r\), and this paper considers the case \(a_1 = 0\). Theorem. Let \(\Gamma\) be a distance-regular graph with \(r = l(1,0,k - 1) > 0\). If \(c_{2r + 1} = 1\) then \(\Gamma\) contains a Moore subgraph of valency \(a_{r + 1} + 1\) and diameter \(r + 1\) (in particular, \(r = 1\) or \(a_{r + 1} = 1)\).