The $\text{bk}(G)$ of a graph $G$ is the maximum number of triangles sharing a common edge. Motivated by a classical conjecture of Erd\H{o}s, spectral lower bounds for the booksize have received considerable attention. For a positive divisor $s$ of $m-1$ with $\frac{m-1}{s}\ge2$, let $S_{m,s}^{+}$ be obtained from $K_{s,\frac{m-1}{s}}$ by adding one edge inside the part of order $\frac{m-1}{s}$. Zhai et al. proved that, apart from this explicit family, every $m$-edge non-bipartite graph satisfying $\rho(G)^2\ge m-1+\frac{2}{\rho(G)-1}$ has booksize greater than $\frac{1}{240}\sqrt{m}$, and they asked for the best possible constant. We answer this question asymptotically. For every $0<\varepsilon<\frac{1}{4}$ and all sufficiently large $m$, every $m$-edge non-bipartite graph $G$ without isolated vertices satisfying the same spectral condition either is isomorphic to $S_{m,s}^{+}$ for some such integer $s$, or satisfies $\text{bk}(G)>\left(\frac{1}{4}-\varepsilon\right)\sqrt{m}$. We also give infinitely many graphs outside the exceptional family showing that no constant larger than $\frac{1}{4}$ is possible. Thus $\frac{1}{4}$ is the optimal asymptotic constant in the problem of Zhai et al.
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.
