On a distance-regular graph with an intersection array {35, 28, 6; 1, 2, 30}

Abstract

It is proved that for a distance-regular graph Γ of diameter 3 with eigenvalue θ2 = -1 the complement graph of Γ3 is pseudo-geometric for pGc3 (k, b1/c2). Bang and Kooien investigated distance-regular graphs with intersection arrays (t + 1)s, ts, (s + 1 - ψ); 1, 2, (t + 1)ψ. If t = 4, s = 7, ψ = 6 then we have array 35, 28, 6; 1, 2, 30. Distance-regular graph Γ with intersection array {35, 28, 6; 1, 2, 30} has spectrum of 351, 9168, -1182, -5273, v = 1 + 35 + 490 + 98 = 624 vertices and Γ3 is a pseudogeometric graph for pG30 (35, 14). Due to the border of Delsarte, the order of clicks in Γ is not more than 8. It is also proved that either a neighborhood of any vertex in Γ is the union of an isolated 7-click, or the neighborhood of any vertex in Γ does not contain a 7-click and is a connected graph. The structure of the group G of automorphisms of a graph Γ with an intersection array {35, 28, 6; 1, 2, 30} has been studied. In particular, π(G) ⊆ {2,3,5,7,13} and the edge symmetric graph Γ has a solvable group automorphisms. © 2019. Southern Mathematical Institute of VSC RAS. All right reserved.