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Let $G$ be an $r$-regular graph of order $n$ and
independence number $\alpha(G)$. We show that if $G$ has odd girth
$2k+3$ then $\alpha(G)\geq n^{1-1/k}r^{1/k}$. We also prove similar
results for graphs which are not regular. Using these results we
improve on the lower bound of Monien and Speckenmeyer, for the
independence number of a graph of order $n$ and odd girth $2k+3$.
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