Merging the A- and Q-spectral theories

Let $G$ be a graph with adjacency matrix $A\left( G\right)$, and let $D\left( G\right)$ be the diagonal matrix of the degrees of $G.$ The signless Laplacian $Q\left( G\right)$ of $G$ is defined as $Q\left( G\right) :=A\left( G\right) +D\left( G\right)$. Cvetkovi\'{c} called the study of the adjacency matrix the $A$% \textit{-spectral theory}, and the study of the signless Laplacian--the $Q$\textit{-spectral theory}. During the years many similarities and differences between these two theories have been established. To track the gradual change of $A\left( G\right)$ into $Q\left( G\right)$ in this paper it is suggested to study the convex linear combinations $A_{\alpha }\left( G\right)$ of $A\left( G\right)$ and $D\left( G\right)$ defined by $$A_{\alpha}\left( G\right) :=\alpha D\left( G\right) +\left( 1-\alpha\right) A\left( G\right) \text{, \ \ }0\leq\alpha\leq1.$$ This study sheds new light on $A\left( G\right)$ and $Q\left( G\right)$, and yields some surprises, in particular, a novel spectral Tur\'{a}n theorem. A number of challenging open problems are discussed.