There are finitely many triangle-free distance-regular graphs with degree 8, 9 or 10
Bannai and Ito conjectured that there are finitely many distance-regular graphs with degree at least 3. This conjecture holds for degrees 3 and 4 (Bannai, Ito), 5, 6 and 7 (Koolen, Moulton). Let \(\Gamma\) be a distance-regular graph with valency \(k\geq 3\) and diameter \(d\geq 2\). Given integers \(a\geq 0\), \(b,c\geq 1\) and \(a+b+c=k\), we define \(l_{(c,a,b)}=\{i\;|\;1\leq i\leq d-1,\;(c_i,a_i,b_i)=(c,a,b)\}\), and put \(h_\Gamma=l_{(1,a_1,b_1)}\) and \(t_\Gamma=l_{(b_1,a_1,1)}\). Theorem. Suppose \(k\geq 3\) is an integer. Then there exists a real number \(\alpha>0\), depending only on \(k\) so that there are finitely many triangle-free distance-regular graphs with degree \(k\) and diameter \(d\) satisfying \(d-(h_\Gamma+t_\Gamma)\leq \alpha h_\Gamma\). Corollary 1.5. There are finitely many triangle-free distance-regular graphs with degree 8, 9, or 10.
scientific article; zbMATH DE number 2082004
scientific article