The $\alpha$-spectral Tur\'an type problems for graphs

For $0 \leq \alpha < 1$, the $\alpha$-spectral radius of a graph $G$ is defined as the largest eigenvalue of $A_{\alpha}(G)=\alpha D(G)+(1-\alpha)A(G)$, where $D(G)$ and $A(G)$ are the diagonal matrix of degrees and adjacency matrix of $G$, respectively. A graph is called color-critical if it contains an edge whose deletion reduces its chromatic number. The celebrated Erd\H{o}s-Stone-Simonovits theorem asserts that $\mathrm{ex}(n,\mathcal{F})=\left(1-\frac{1}{\chi(\mathcal{F})-1}+o(1)\right)\frac{n^2}{2},$ where $\chi(\mathcal{F})$ is the chromatic number of $\mathcal{F}$. Nikiforov and Zheng et al. established the adjacency spectral and signless Laplacian spectral versions of this theorem, respectively. In this paper, we present the $\alpha$-spectral version of this theorem, which unifies the aforementioned results. Furthermore, we characterize the $\alpha$-spectral extremal graphs for color-critical graphs, thereby extending the existing results on adjacency spectral and signless Laplacian spectral extremal graphs for such graphs.