A characterization of the odd graphs and the doubled odd graphs with a few of their intersection numbers (Q854827)

The main result is the following: Let \(\Gamma \) be a distance regular graph of diameter \(d\), valency \(k\geq 4\) and \(r=| \{ i\mid (c_i,a_i,b_i) = (c_1,a_1,b_1)\} |\). Suppose one of the following conditions holds. (i) There exists an integer \(m\) with \(r\leq m\leq d-r-1\) such that \(a_{m+r}=0 \) and \(1+c_{m}=c_{m+r}\leq k-2\). (ii) There exists an integer \(m\) with \(r\leq m\leq d-r-1\) such that \(a_{m}=0\) and \(2\leq b_{m+r}=b_{m}-1\). (iii) There exist integers \(m\) and \(t\) with \(2r\leq m\leq d-1\) and \(r\leq t\leq m-r\) such that \(c_{m-1}<c_{m}\), \(a_{1}=\cdots =a_{m}=0\) and \( c_{t}+c_{m-t}=c_{m}\). (iv) There exist integers \(m\) and \(t\) with \(2r\leq m\leq d-r\) and \(r\leq t\leq m-r\) such that \(c_{t}<c_{t+1}\), \(a_{1}=\cdots =a_{m}=0\) and \( c_{t}+c_{m-t}=c_{m}\). Then \(\Gamma \) is either the odd graph, or the double odd graph.