Non-trivial ΓΔ-regular graphs

Abstract

Using the results of C.D. Godsil and B.D. McKay in “Graphs with regular neighbourhoods” we prove that there are only two non-trivial ΓΔ-regular graphs with diameter 3, and that all the other non-trivial ΓΔ-regular graphs have diameter 2. We also prove that there are no non-trivial ΓΔ-regular graphs with η> λ+1. Next we provethat if a (0,1)-geometric graph is a non-trivial ΓΔ-regular graph, it is one of the two non-trivial ΓΔ-regular graphs with diameter 3. At last we construct two new non-trivial ΓΔ-regular graphs on 28 vertices using the Chang graphs.