A class of bipartite and antipodal graphs and their uniform posets

Abstract

Let \( \Gamma \) denote a bipartite and antipodal distance-regular graph with vertex set \( X \), diameter \( D \) and valency \( k \). Firstly, we determine such graphs \( \Gamma \) when \( D \geq 8 \), \( k \geq 3 \) and their corresponding quotient graphs are \( Q \)-polynomial: \( \Gamma \) is a \( 2d \)-cube if \( D = 2d \); \( \Gamma \) is either a \( (2d+1) \)-cube or the doubled Odd graph if \( D = 2d+1 \). Secondly, by defining a partial order \( \leq \) on \( X \) we obtain a grading poset \( (X, \leq) \) 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 \( (X, \leq) \) for \( D \)-cube is uniform. In this paper, we prove that the poset \( (X, \leq) \) for the doubled Odd graph is not uniform.