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 $G\mathrm{.}$ 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 $A$% $\mathit{\text{-spectral}}\mathit{\text{theory}}$, and the study of the signless Laplacian–the $Q$$\mathit{\text{-spectral}}\mathit{\text{theory}}$. During the years many similarities and differences between these two theories have been established. To track the gradual change of $A\left(G\right)$ into $Q\left(G\right)$ in this paper it is suggested to study the convex linear combinations $A_{\alpha }\left( G\right) $of$A\left( G\right) $and$D\left( G\right) $ defined by $$A_{\alpha }\left(G\right):=\alpha D\left(G\right)+(1-\alpha )A\left(G\right)\text{, }0\le \alpha \le 1.$$ 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.