Spectral Radius of Graphs with Given Size and Odd Girth

Let G(m,k) be the set of graphs with size m and odd girth (the length of shortest odd cycle) k. In this paper, we determine the graph maximizing the spectral radius among G(m,k) when m is odd. As byproducts, we show that, there is a number η(m,k)>√m-k+3 such that every non bipartite graph G with size m and spectral radius ρ≥ η(m,k) must contain an odd cycle of length less than k unless m is odd and G SKk,m, which is the graph obtained by subdividing an edge $k-2$ times of the complete bipartite graph K2,m-k+2/2. This result implies the main results of Zhai and Shu [Discrete Math. 345 (2022)] and settles a conjecture of Li and Peng [The Electronic J. Combin. 29 (4) (2022)] as well.