Twice $Q$-polynomial distance-regular graphs of diameter 4

Abstract

It is known that a distance-regular graph with valency $k$ at least three admits at most two Q-polynomial structures. % In this note we show that all distance-regular graphs with diameter four and valency at least three admitting two $Q$-polynomial structures are either dual bipartite or almost dual imprimitive. By the work of Dickie \cite{Dickie} this implies that any distance-regular graph with diameter $d$ at least four and valency at least three admitting two $Q$-polynomial structures is, provided it is not a Hadamard graph, either the cube $H(d,2)$ with $d$ even, the half cube $1/2H(2d+1,2)$, the folded cube $\tilde{H}(2d+1,2)$, or the dual polar graph on ${[}^2{A}_{2d-1}(q)]$ with $q\ge 2$ a prime power.

8 pages

References in corpus (2)

• Distance-regular graphs
• Nonexistence of exceptional imprimitive Q-polynomial association schemes with six classes

Cited by in corpus (1)

• Distance-regular graphs