Determine the exact maximum in nonzero residue classes for odd-wheel-free graphs

Determine the exact maximum spectral radius of a $W_{2k+1}$-free graph with a sufficiently large fixed number $m$ of edges when $k\geq3$ and $m-inom{k}{2}$ is not divisible by $k$, and characterize the extremal graphs, with particular attention to threshold graphs obtained from $K_k\vee qK_1$ by introducing a bounded defect.

Background

Theorem 2 identifies the equality case for the Brualdi--Hoffman--Turán spectral problem for $W_{2k+1}$-free graphs, $k\ge3$, only when the edge count has the form $m=\binom{k}{2}+kq$. Consequently, equality is impossible in the other residue classes modulo $k$. The concluding remarks identify determining the exact extremal value in these remaining classes as a finer fixed-size problem and indicate threshold graphs with bounded defects as natural candidates.