$A_{\alpha }$ matrix of commuting graphs of non-abelian groups

For a finite group $\mathcal{G}$ and a subset $X\ne \mathrm{\varnothing }$ of $\mathcal{G}$, the commuting graph, indicated by $G=\mathcal{C}(\mathcal{G},X)$, is the simple connected graph with vertex set $X$ and two distinct vertices $x$ and $y$ are edge connected in $G$ if and only if they commute in $X$. The $A_{\alpha }$ matrix of $G$ is specified as $A_{\alpha }(G)=\alpha D(G)+(1-\alpha )A(G),\alpha \in [0,1]$, where $D(G)$ is the diagonal matrix of vertex degrees while $A(G)$ is the adjacency matrix of $G.$ In this article, we investigate the $A_{\alpha }$ matrix for commuting graphs of finite groups and we also find the $A_{\alpha }$ eigenvalues of the dihedral, the semidihedral and the dicyclic groups. We determine the upper bounds for the largest $A_{\alpha }$ eigenvalue for these graphs. Consequently, we get the adjacency eigenvalues, the Laplacian eigenvalues, and the signless Laplacian eigenvalues of these graphs for particular values of $\alpha$. Further, we show that these graphs are Laplacian integral.