A spectral condition for the existence of cycles with consecutive odd lengths in non-bipartite graphs

A graph G is called H-free, if it does not contain H as a subgraph. In 2010, Nikiforov proposed Brualdi-Solheid-Turán type problem: what the maximum spectral radius of an H-free order n? this paper, we consider problem for non-bipartite graphs. Let Ka,b•K3 denote obtained by identifying vertex Ka,b in part size b and K3. We prove that n satisfying ρ(G)≥ρ(K⌈n−22⌉,⌊n−22⌋•K3), then contains all odd cycles C2l+1 each integer l∈[2,k] unless G≅K⌈n−22⌉,⌊n−22⌋•K3, provided sufficiently large with respect to k. This resolves posed Guo, Lin Zhao (2021).