The parameters of bipartite $Q$-polynomial distance-regular graphs

Caughman, John S.IV

Journal of Algebraic Combinatorics, Tome 15 (2002) no. 3, pp. 223-229.

Résumé

Summary: Let $Gamma$ denote a bipartite distance-regular graph with diameter $Dge$ 3 and valency $kge$ 3. Suppose $theta _{0}, theta _{1}, \dots , theta_{ D }$ is a $Q$-polynomial ordering of the eigenvalues of $Gamma$. This sequence is known to satisfy the recurrence $theta_{ i - 1} - betatheta_{ i } + theta_{ i + 1} = 0 (0 > i > D)$, for some real scalar $beta$. Let $q$ denote a complex scalar such that $q + q ^{-1} = beta$. Bannai and Ito have conjectured that $q$ is real if the diameter $D$ is sufficiently large. We settle this conjecture in the bipartite case by showing that $q$ is real if the diameter $Dge$ 4. Moreover, if $D = 3$, then $q$ is not real if and only if $theta _{1}$ is the second largest eigenvalue and the pair ( $mgr, k$) is one of the following: (1, 3), (1, 4), (1, 5), (1, 6), (2, 4), or (2, 5). We observe that each of these pairs has a unique realization by a known bipartite distance-regular graph of diameter 3.

Détail

Keywords: distance-regular graph, bipartite, association scheme, $P$-polynomial, $Q$-polynomial