We present a unified presentation of the fixed-size adjacency-spectral extremal problem for odd wheels $W_{2k+1}$, where $k\geq2$ and $W_{2k+1}=K_1\vee C_{2k}$. The exceptional case $W_5$ and the general case $W_{2k+1}$, $k\ge3$, share the same dense-core reduction and edge-spectral stability, but have different rigidity structures. We prove that every $W_5$-free graph of sufficiently large size $m$ satisfies $\rho(G)^2-\rho(G)\le m,$ with equality precisely for $K_{n,n}$ with a perfect matching embedded in each part, where $n$ is even and $m=n^2+n$. For any fixed $k\ge3$, every $W_{2k+1}$-free graph of sufficiently large size $m$ satisfies $\rho(G)^2-(k-1)\rho(G)\le m-\binom{k}{2},$ with equality precisely for $K_k\vee qK_1$ when $m=\binom{k}{2}+kq$. Our results completely settle a conjecture proposed by Yu, Li and Peng and, via a distinct approach, further strengthen known results concerning odd cycles, friendship graphs and odd fan graphs for sufficiently large $m.$ The proof combines the edge-spectral stability theorem, residual functions and the dense-core method.
A central theme in extremal graph theory is the supersaturation problem, which investigates the minimum number of copies of a target subgraph forced by prescribed edge conditions. This line of research goes back to Rademacher and Erd\H{o}s for triangles, and was later extended to cliques by Lov\'asz and Simonovits in the regime above the Tur\'an threshold. Mubayi further extended this theory to color-critical graphs. Below the Tur\'an threshold, a closely related existence-threshold phenomenon arises in the non-$p$-partite setting: a classical result of Brouwer shows that, for $n\ge 2p+1$, every $n$-vertex non-$p$-partite $K_{p+1}$-free graph has at most $e(T_{n,p})-\lfloor n/p\rfloor+1$ edges. Motivated by this threshold, we investigate a sharp clique-counting problem below the Tur\'an threshold under the non-$p$-partite assumption. Let $p\ge 2$ and $s\ge 1$ be fixed integers. Let $Y_{n,p,s}$ be the graph obtained from $T_{n,p}$ by adding an edge inside a largest part and deleting all but $s$ of the edges from one endpoint of this new edge to a smallest part. Then $e(Y_{n,p,s})=e(T_{n,p})-\lfloor n/p\rfloor+s+1$. We prove that, for all sufficiently large $n$, every $n$-vertex non-$p$-partite graph $G$ with $e(G)\ge e(Y_{n,p,s})$ contains at least as many copies of $K_{p+1}$ as $Y_{n,p,s}$ does. The bound is sharp, as it is attained by the construction $Y_{n,p,s}$. Thus our result provides the exact clique-counting analogue of Brouwer's threshold for non-$p$-partite $K_{p+1}$-free graphs.
