Maximum spectral radius conjecture for odd-cycle-free non-bipartite graphs

Let $G$ be a graph with $m$ edges. For integers $k\ge 1$, let $S_{2k-1}(K_{s,t})$ be obtained from the complete bipartite graph $K_{s,t}$ by replacing an edge with a path $P_{2k+1}$, introducing $2k-1$ new vertices. Its odd girth is $2k+3$.

Maximum spectral radius conjecture. If $G$ contains no member of $\{C_3,C_5,\ldots,C_{2k+1}\}$ and is non-bipartite, then

$$\lambda(G)\leq \lambda\left(S_{2k-1}\left(K_{2,\frac{m-2k+1}{2}}\right)\right),$$

with equality if and only if $m$ is odd and $G=S_{2k-1}\left(K_{2,\frac{m-2k+1}{2}}\right)$.

The conjecture proposes the extremal non-bipartite graph for the largest spectral radius under the prescribed odd-cycle exclusions; the paper notes that its methods support the claim for progressively longer forbidden odd cycles, but the general assertion remains open.