  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.
