  In this paper we show that for an invariant $(\alpha,\beta)-$metric $F$ on a
homogeneous Finsler manifold $\frac{G}{H}$, induced by an invariant Riemannian
metric $\tilde{a}$ and an invariant vector field $\tilde{X}$, the vector
$X=\tilde{X}(H)$ is a geodesic vector of $F$ if and only if it is a geodesic
vector of $\tilde{a}$. Then we give some conditions such that under them, an
arbitrary vector is a geodesic vector of $F$ if and only if it is a geodesic
vector of $\tilde{a}$. Finally we give an explicit formula for the flag
curvature of bi-invariant $(\alpha,\beta)-$metrics on connected Lie groups.
