On bipartite Q-polynomial distance-regular graphs with $c_2\le 2$

Safet Penjić
University of Primorska

Štefko Miklavič
University of Primorska

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Minisymposium: ASSOCIATION SCHEMES

Content: Let $\Gamma$ denote a $Q$-polynomial bipartite distance-regular graph with diameter $D$, valency $k \ge 3$ and intersection number $c_2\le 2$. In this talk we show the following two results: (1) If $D \ge 6$, then $\Gamma$ is either the $D$-dimensional hypercube, or the antipodal quotient of the $2D$-dimensional hypercube. (2) If $D = 4$ then $\Gamma$ is either the $4$-dimensional hypercube, or the antipodal quotient of the $8$-dimensional hypercube. We show (1) using results of Caughman. To show (2) we first introduce certain equitable partition of the vertex-set of $\Gamma$. Then we use this equitable partition to prove (2).

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