On small distance-regular graphs with the intersection arrays {mn − 1, (m − 1)(n + 1), n − m + 1; 1, 1, (m − 1)(n + 1)}

Abstract

Let Γ be a diameter 3 distance-regular graph with a strongly regular graph Γ₃, where Γ₃ is the graph whose vertex set coincides with the vertex set of the graph Γ and two vertices are adjacent whenever they are at distance 3 in the graph Γ. Computing the parameters of Γ₃ by the intersection array of the graph Γ is considered as the direct problem. Recovering the intersection array of the graph Γ by the parameters of Γ₃ is referred to as the inverse problem. The inverse problem for Γ₃ has been solved earlier by A. A. Makhnev and M. S. Nirova. In the case where Γ₃ is a pseudo-geometric graph of a net, a series of admissible intersection arrays has been obtained: {c₂(u² − m²) + 2c₂m − c₂ − 1, c₂(u² − m²), (c₂ − 1)(u² − m²) + 2c₂m − c₂; 1, c₂, u² − m²} (A. A. Makhnev, Wenbin Guo, M. P. Golubyatnikov). The cases c₂ = 1 and c₂ = 2 have been examined by A. A. Makhnev, M. P. Golubyatnikov and A. A. Makhnev, M. S. Nirova, respectively.

In this paper in the class of graphs with the intersection arrays {mn − 1, (m − 1)(n + 1)}, {n − m + 1}; 1, 1, (m − 1)(n + 1)} all admissible intersection arrays for {3 ≤ m ≤ 13} are found: {20,16,5; 1, 1,16}, {39,36,4; 1, 1,36}, {55,54,2; 1, 2,54}, {90,84,7; 1, 1,84}, {220,216,5; 1, 1,216}, {272,264,9; 1, 1,264} and {350,336,15; 1, 1,336}. It is demonstrated that graphs with the intersection arrays {20,16,5; 1, 1,16}, {39,36,4; 1, 1,36} and {90,84,7; 1, 1,84} do not exist.