Abstract: Let $\Gamma$ be a nonbipartite distance-regular
graph with $d\geq 3$, and eigenvalues
$\theta_{0}>\theta_{1}>\cdots >\theta_{d}.$ Let $\theta_{1'}\in\{ \theta_{1},\theta_{d}\},$
and let $\theta_{d'}$ be the complement of $\theta_{1'}$ in
$\{ \theta_{1},\theta_{d}\}$. Assume that $\Gamma$ is of type
$E_{1}\circ E_{d}$ with property $E_{1}\circ E_{d}=|X|^{-1}(q^{d-1}_{1d}E_{d-1}+q^{d}_{1d}E_{d})$ and that
$\sigma_{0},\sigma_{1},\cdots,\sigma_{d}$,
$\rho_{0},\rho_{1},\cdots,\rho_{d}$ and
$\beta_{0},\beta_{1},\cdots,\beta_{d}$ are the cosine sequences
associated with $\theta_{1'}$,$\theta_{d'}$ and
$\theta_{d-1}$, respectively. Using the above cosine sequences, the authors
give some equivalence conditions of $\Gamma$ being $Q$-polynomial
with respect to $\theta_{1}$ or $\theta_{d}.$

Key words: Distance-regular graphs, Cosine sequences, Q -polynomial, Krein parameters