2024/04/01 by Koolen, Jack H., Abdullah, Mamoon, Gebremichel, Brhane +1 · 1 citation
#05C50 #05E30 #Combinatorics (math.CO) #FOS: Mathematics
paper · doi:10.48550/arxiv.2404.01134
Let Γ be a graph with diameter at least two. Then Γ is said to be 1-homogeneous (in the sense of Nomura) whenever for every pair of adjacent vertices x and y in Γ, the distance partition of the vertex set of Γ with respect to both x and y is equitable, and the parameters corresponding to equitable partitions are independent of the choice of x and y. Assume that Γ is 1-homogeneous distance-regular with intersection number a1>0 and diameter D\geqslant 5. Define b=b1/(θ1+1), where b1 is the intersection number and θ1 is the second largest eigenvalue of Γ. We show that if intersection number c2 is at least 2, then b\geqslant 1 and one of the following (i)--(vi) holds: (i) Γ is a regular near 2D-gon, (ii) Γ is a Johnson graph J(2D,D), (iii) Γ is a halved ℓ-cube with ℓ ∈ \2D,2D+1\, (iv) Γ is a folded Johnson graph J(4D,2D), (v) Γ is a folded halved 4D-cube, (vi) the valency of Γ is bounded by a function of b. Using this result, we characterize 1-homogeneous graphs with classical parameters and a1>0, as well as tight distance-regular graphs.