Bipartite Q-polynomial distance-regular graphs and uniform posets [article]

Stefko Miklavic, Paul Terwilliger
2011 arXiv   pre-print
Let denote a bipartite distance-regular graph with vertex set X and diameter D > 3. Fix x ∈ X and let L (resp. R) denote the corresponding lowering (resp. raising) matrix. We show that each Q-polynomial structure for yields a certain linear dependency among RL^2, LRL, L^2R, L. Define a partial order < on X as follows. For y,z ∈ X let y < z whenever ∂(x,y)+∂(y,z)=∂(x,z), where ∂ denotes path-length distance. We determine whether the above linear dependency gives this poset a uniform or strongly
more » ... niform structure. We show that except for one special case a uniform structure is attained, and except for three special cases a strongly uniform structure is attained.
arXiv:1108.2484v1 fatcat:tfxrnwpntzfqdmze63wbrrsysi