1-homogeneous, pseudo-1-homogeneous, and 1-thin distance-regular graphs
Let \(\Gamma\) be a distance-regular graph with diameter \(d\) and fix a vertex \(x\) of \(\Gamma\). \(\Gamma\) is said to be 1-homogeneous (resp. pseudo-1-homogeneous) with respect to \(x\) whenever for all integers \(h\) and \(i\) between 0 and \(d\) (resp. for all integers \(h\) between 0 and \(d-1\) and \(i\) between 0 and \(d\)) and for all vertices \(y,z\) of \(\Gamma\) with \(d(x,y)=h\), \(d(y,z)=i\), \(d(z,x)=1\), the number \(\gamma^{h,i}_{j,k}\) of vertices \(w\) of \(\Gamma\) with \(d(x,w)=j\), \(d(y,w)=1\), \(d(z,w)=k\) is independent of \(y\) and \(z\) for all \(j\) and \(k\) between 0 and \(d\). Theorem 1.1. Let \(\Gamma\) be a distance-regular graph with diameter \(d\). Fix a base vertex \(x\) of \(\Gamma\), and write \(E_i^*=E_i^*(x)\), \({\mathcal M}^*= {\mathcal M}^*(x)\) and \({\mathcal T}={\mathcal T}(x)\). Then the following conditions are equivalent. (i) \(\Gamma\) is 1-homogeneous with respect to \(x\). (ii) \({\mathcal T}E_1^*={\mathcal M}^*{\mathcal M}E_1^*\). (iii) \(\dim E_i^*{\mathcal T}E_1^*\leq 3\) (\(1\leq i\leq d-1\)) and \(\dim E_d^*{\mathcal T}E_1^*\leq 2\). Theorem 1.4. Let \(\Gamma\) be a distance-regular graph with diameter \(d\geq 2\). Fix a base vertex \(x\) of \(\Gamma\), and write \(E_i^*=E_i^*(x)\), \({\mathcal M}^*= {\mathcal M}^*(x)\) and \({\mathcal T}={\mathcal T}(x)\). Assume that \(a_1\neq 0\). Then the following conditions are equivalent. (i) \(\Gamma\) is pseudo-1-homogeneous with respect to \(x\). (ii) \({\mathcal T}E_1^*\) is spanned by \(\{E_0^*AE_1^*\}\cup \{E_i^*A_{i-1}E_1^*,E_i^*A_iE_1^*,E_i^*A_{i+1}E_1^*\;|\;1\leq i\leq d-1\}\cup \{E_d^*A_{d-1}E_1^*,E_d^*A_dE_1^*,E_d^*AE_d^*A_{d-1}E_1^*\}\). (iii) \(\dim E_i^*{\mathcal T}E_1^*\leq 3\) (\(1\leq i\leq d\)).
scientific article; zbMATH DE number 2142251