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

TL;DR

This paper characterizes distance-regular graphs with diameter four and two Q-polynomial structures, showing they are either dual bipartite or almost dual imprimitive, and classifies such graphs under certain conditions.

Contribution

It proves that all diameter four, valency at least three, two Q-polynomial structure graphs are either dual bipartite or almost dual imprimitive, extending previous classifications.

Findings

Graphs are either dual bipartite or almost dual imprimitive.

Classification includes cubes, half cubes, folded cubes, and dual polar graphs.

Results extend to graphs with diameter at least four, excluding Hadamard graphs.

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 ${1}/{2} H(2d+1,2)$, the folded cube $\tilde{H}(2d+1,2)$, or the dual polar graph on $[^2A_{2d-1}(q)]$ with $q\ge 2$ a prime power.