TY - JOUR
T1 - A note on distance-regular graphs with a small number of vertices compared to the valency
AU - Koolen, Jack H.
AU - Park, Jongyook
PY - 2013/8
Y1 - 2013/8
N2 - In this note we study distance-regular graphs with a small number of vertices compared to the valency. We show that for a given α > 2, there are finitely many distance-regular graphs Γ with valency k, diameter D ≥ 3 and v vertices satisfying v≤αk unless (D = 3 and Γ is imprimitive) or (D = 4 and Γ is antipodal and bipartite). We also show, as a consequence of this result, that there are finitely many distance-regular graphs with valency k ≥ 3, diameter D ≥ 3 and c2 ≥ εk for a given 0 < ε < 1 unless (D = 3 and Γ is imprimitive) or (D = 4 and Γ is antipodal and bipartite).
AB - In this note we study distance-regular graphs with a small number of vertices compared to the valency. We show that for a given α > 2, there are finitely many distance-regular graphs Γ with valency k, diameter D ≥ 3 and v vertices satisfying v≤αk unless (D = 3 and Γ is imprimitive) or (D = 4 and Γ is antipodal and bipartite). We also show, as a consequence of this result, that there are finitely many distance-regular graphs with valency k ≥ 3, diameter D ≥ 3 and c2 ≥ εk for a given 0 < ε < 1 unless (D = 3 and Γ is imprimitive) or (D = 4 and Γ is antipodal and bipartite).
UR - http://www.scopus.com/inward/record.url?scp=84874400296&partnerID=8YFLogxK
U2 - 10.1016/j.ejc.2013.01.006
DO - 10.1016/j.ejc.2013.01.006
M3 - Article
AN - SCOPUS:84874400296
SN - 0195-6698
VL - 34
SP - 935
EP - 940
JO - European Journal of Combinatorics
JF - European Journal of Combinatorics
IS - 6
ER -