Triangle-free distance-regular graphs

A Higman inequality for regular near polygons

“…Weng [39] proved that distanceregular graphs with classical parameters (d, b, α, β) with b < −1, d ≥ 4, c 2 > 1, and with triangles must either be in one of two known families or satisfy (d, b, α, β) = (d, −q, −(q + 1)/2, −((−q) d + 1)/2) for some odd prime power q. For q = 3, the induced graph on a 2-ovoid in the dual polar graph from H (2d − 1, q 2 ) would be a triangle-free distance-regular graph with classical parameters (d, b, α, β) = (d, −3, −2, −((−3) d + 1)/2), the nonexistence of which was conjectured for d ≥ 3 in [27,Conjecture 4.11].…”

Nonexistence of a Class of Distance-Regular Graphs

“…Throughout this section, let Γ = (X, R) denote a distance-regular graph with diameter D 3, and intersection numbers a 1 = 0, a 2 = 0. Such graphs are also studied in [4,5,6,7,8].…”

“…We show a connection between the d-bounded property and the nonexistence of parallelograms of any length up to d + 1. Assume further that Γ is with classical parameters (D, b, α, β), Pan and Weng (2009) showed that (b, α, β) = (−2, −2, ((−2) D+1 −1)/3). Under the assumption D 4, we exclude this class of graphs by an application of the above connection.…”

3-bounded property in a triangle-free distance-regular graph

“…Note that if Γ has classical parameters (D, b, α, β) with D ≥ 3, a 1 = 0 and a 2 = 0, then Γ contains no parallelograms of any length. See [6,Theorem 1.1] or Theorem 3.3 in this article.…”

“…([6, Theorem 1.1]) Let Γ denote a distance-regular graph with diameter D ≥ 3 and intersection numbers a 1 = 0, a 2 = 0. Then the following (i)-(iii) are equivalent.…”