  It is well known that spectral Tur\'{a}n type problem is one of the most
classical {problems} in graph theory. In this paper, we consider the spectral
Tur\'{a}n type problem. Let $G$ be a graph and let $\mathcal{G}$ be a set of
graphs, we say $G$ is \textit{$\mathcal{G}$-free} if $G$ does not contain any
element of $\mathcal{G}$ as a subgraph. Denote by $\lambda_1$ and $\lambda_2$
the largest and the second largest eigenvalues of the adjacency matrix $A(G)$
of $G,$ respectively. In this paper we focus on the characterization of graphs
without short odd cycles according to the adjacency eigenvalues of the graphs.
Firstly, an upper bound on $\lambda_1^{2k}+\lambda_2^{2k}$ of $n$-vertex
$\{C_3,C_5,\ldots,C_{2k+1}\}$-free graphs is established, where $k$ is a
positive integer. All the corresponding extremal graphs are identified.
Secondly, a sufficient condition for non-bipartite graphs containing an odd
cycle of length at most $2k+1$ in terms of its spectral radius is given. At
last, we characterize the unique graph having the maximum spectral radius among
the set of $n$-vertex non-bipartite graphs with odd girth at least $2k+3,$
which solves an open problem proposed by Lin, Ning and Wu [Eigenvalues and
triangles in graphs, Combin. Probab. Comput. 30 (2) (2021) 258-270].
