Three Infinite Families of Shilla Graphs Do Not Exist

A distance-regular graph of diameter 3 with the second eigenvalue $${{\theta }_{1}} = {{a}_{3}}$$ is called a Shilla graph. For a Shilla graph Γ, the number a = a3 divides k and we set $$b = b(\Gamma ) = k{\text{/}}a$$ . Three infinite families of Shilla graphs with the following admissible intersection arrays were found earlier: {b(b2 – 1), b2(b – 1), b2; 1, 1, $$({{b}^{2}} - 1)(b - 1)\} $$ (I.N. Belousov), $$\{ {{b}^{2}}(b - 1){\text{/}}2,(b - 1)({{b}^{2}} - b + 2){\text{/}}2$$ , $$b(b - 1){\text{/}}4;$$ $$1,b(b - 1){\text{/}}4,b{{(b - 1)}^{2}}{\text{/}}2\} $$ (Koolen, Park), and {(s + 1) $$({{s}^{3}} - 1),{{s}^{4}},{{s}^{3}};1,{{s}^{2}},s({{s}^{3}} - 1)\} $$ (Belousov). In this paper, it is proved that, in the first family, there exists a unique graph, namely, a generalized hexagon of order 2, whereas there are no graphs in the second or third families.