Search

A non-complete distance-regular graph is called geometric if there exists a set C of Delsarte cliques such that each edge lies in exactly one clique in C . Let Γ be a geometric distance-regular graph with diameter D ≥ 3 and smallest eigenvalue θ D . In this paper we show that if Γ contains an induced subgraph K 2 , 1 , 1 , then D ≤ − θ D . Moreover, if − θ D − 1 ≤ D ≤ − θ D then D = − θ D and Γ is a Johnson graph. We also show that for ( s , b ) ⁄ ∈ { ( 11 , 11 ) , ( 21 , 21 ) } , there are no distance-regular graphs with intersection array { 4 s , 3 ( s − 1 ) , s + 1 − b ; 1 , 6 , 4 b } where s , b are integers satisfying s ≥ 3 and 2 ≤ b ≤ s . As an application of these results, we classify geometric distance-regular graphs with D ≥ 3 , θ D ≥ − 4 and containing an induced subgraph K 2 , 1 , 1 . [ABSTRACT FROM AUTHOR]

For a distance-regular graph of diameter three Γ, the statement that distance-3 graph Γ 3 of Γ is strongly regular is equivalent to that Γ has eigenvalue −1. There are many distance-regular graphs of diameter 3 having eigenvalue −1, such as the folded 7-cube, generalized hexagons of order ( s , s ) and antipodal nonbipartite distance-regular graphs of diameter 3. In this paper, we show that for a fixed positive integer α ( β , respectively), there are only finitely many distance-regular graphs of diameter 3 having eigenvalue −1 and a 3 = α ( b 1 c 2 = β and a 3 ≠ 0 , respectively). Such distance-regular graphs for small numbers α = 1 , 2 or β = 3 with a 3 ≠ 0 are classified. We show that there are no distance-regular graphs with intersection array { 44 , 35 , 3 ; 1 , 5 , 42 } . Moreover, we classify the distance-regular graphs with diameter 3 and smallest eigenvalue greater than −3. [ABSTRACT FROM AUTHOR]