Adjacency eigenvalues of graphs without short odd cycles

It is well known that spectral Turán type problem one of the most classical problems in graph theory. In this paper, we consider problem. Let G be a and let set graphs, say G-free if does not contain any element as subgraph. Denote by ?1 ?2 largest second eigenvalues adjacency matrix A(G) G, respectively. paper focus on characterization graphs without short odd cycles according to graphs. Firstly, an upper bound ?12k+?22k n-vertex {C3,C5,…,C2k+1}-free established, where k positive integer. All corresponding extremal are identified. Secondly, sufficient condition for non-bipartite containing cycle length at 2k+1 terms its radius given. At last, characterize unique having maximum among with girth least 2k+3, which solves open proposed Lin, Ning Wu [Eigenvalues triangles Combin. Probab. Comput. 30 (2) (2021) 258-270].