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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