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 (m, k) >m-k+3 such that every non bipartite graph G with size m and spectral radius (m, k) must contain an odd cycle of length less than k unless m is odd and G SK₊, ₌, which is the graph obtained by subdividing an edge k-2 times of the complete bipartite graph K₂, ₌-₊+₂₂. 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.