  A new method is introduced for bounding the separation between the value of
$-k$ and the smallest eigenvalue of a non-bipartite $k$-regular graph. The
method is based on fractional decompositions of graphs. As a consequence we
obtain a very short proof of a generalization and strengthening of a recent
result of Qiao, Jing, and Koolen [Non-bipartite distance-regular graphs with a
small smallest eigenvalue, Electronic J. Combin. 26(2) (2019), P2.41] about the
smallest eigenvalue of non-bipartite distance-regular graphs.
