  It is well-known that eigenvalues of graphs can be used to describe
structural properties and parameters of graphs. A theorem of Nosal states that
if $G$ is a triangle-free graph with $m$ edges, then $\lambda (G)\le \sqrt{m}$,
equality holds if and only if $G$ is a complete bipartite graph. Recently, Lin,
Ning and Wu [Combin. Probab. Comput. 30 (2021)] proved a generalization for
non-bipartite triangle-free graphs. Moreover, Zhai and Shu [Discrete Math. 345
(2022)] presented a further improvement. In this paper, we present an
alternative method for proving the improvement by Zhai and Shu. Furthermore,
the method can allow us to give a refinement on the result of Zhai and Shu for
non-bipartite graphs without short odd cycles.
