Four class association scheme double covers of strongly regular graphs

Source abstract

In this paper we give necessary conditions for 4-class association schemes that are generated by double covers of strongly regular graphs. These conditions are applied to open cases for diameter 4 antipodal distance-regular graphs. Using these conditions we are able to show the nonexistence of four cases in the table of Brouwer, Cohen and Neumeier: $\{20,18,3,1;1,3,18,20\}$, $\{22,21,3,1;1,3,21,22\}$, $\{ 54,50,5,1;,1,5,50,54 \}$, $\{ 170,162,9,1;,1,9,162,170 \}$. More generally, we show there is no distance-regular graph with intersection array $\{k,b_1,b_2,1;1,b_2,b_1,k \}$ where $b_2 \neq 1$ and $\frac{ k b_1 }{4}(1+k+\frac{ k b_1 }{2b_2})$ is odd. Tables are also given for more general 4-class association schemes generated by double covers of strongly regular graphs.