Two new infinite families of arc-transitive antipodal distance-regular graphs of diameter three with $$\lambda =\mu $$ λ = μ related to groups $$Sz(q)$$ S z ( q ) and $$^2G_2(q)$$ 2 G 2 ( q )

Abstract

In this paper, we construct two new infinite families of arc-transitive distance-regular graphs, related to Suzuki groups $$Sz(q)$$Sz(q) and Ree groups $$^2G_2(q)$$2G2(q), where $$q>3$$q>3. They are antipodal $$r$$r-covers of complete graphs on $$q^2+1$$q2+1 or $$q^3+1$$q3+1 vertices, respectively, with $$\lambda =\mu $$?=μ and $$r>1$$r>1 being an arbitrary odd divisor of $$q-1$$q-1. We also find that the graph on the set of involutions of $$Sz(q)$$Sz(q) with $$q>3$$q>3, whose edges are the pairs of involutions $$\{u,v\}$${u,v} such that $$|uv|=5$$|uv|=5, is distance-regular.