20.54 (2022)

A distance-regular graph of diameter 3 with the second eigenvalue $\theta_1 = a_3$ is called a Shilla graph. For a Shilla graph $\Gamma$ the number $a = a_3$ divides $k$ and we set $b = b(\Gamma) = k/a$. Koolen and Park proved that there are 12 feasible intersection arrays of Shilla graphs with $b = 3$. At present it is proved that a Shilla graph with $b = 3$ has intersection array $\{12, 10, 3; 1, 3, 8\}$ (Doro graph), $\{12, 10, 5; 1, 1, 8\}$ (nonisotropic unitary graph for $q = 4$), or $\{15, 12, 6; 1, 2, 10\}$. The automorphisms of the last graph were found by A. Makhnev and N. Zyulyarkina (Doklady Maths., 84, no. 1 (2011), 510–514). Does the graph with intersection array $\{15, 12, 6; 1, 2, 10\}$ exist?