  Recently, it has been shown that a connected graph $\Gamma$ with $d+1$
distinct eigenvalues and odd-girth $2d+1$ is distance-regular. The proof of
this result was based on the spectral excess theorem. In this note we present
an alternative and more direct proof which does not rely on the spectral excess
theorem, but on a known characterization of distance-regular graphs in terms of
the predistance polynomial of degree $d$.
