A class of bipartite and antipodal graphs and their uniform posets

Abstract

Let $\mathrm{\Gamma }$ denote a bipartite and antipodal distance-regular graph with vertex set $X$, diameter $D$ and valency $k$. Firstly, we determine such graphs $\mathrm{\Gamma }$ when $D\ge 8$, $k\ge 3$ and their corresponding quotient graphs are $Q$-polynomial: $\mathrm{\Gamma }$ is a $2d$-cube if $D=2d$; $\mathrm{\Gamma }$ is either a $(2d+1)$-cube or the doubled Odd graph if $D=2d+1$. Secondly, by defining a partial order $\le$ on $X$ we obtain a grading poset $(X,\le )$ with rank $D$. In [Š. Miklavič, P. Terwilliger, Bipartite $Q$-polynomial distance-regular graphs and uniform posets, J. Algebr. Combin. 225-242 (2013)], the authors determined precisely whether the poset $(X,\le )$ for $D$-cube is uniform. In this paper, we prove that the poset $(X,\le )$ for the doubled Odd graph is not uniform.