Let G be a graph with adjacency matrix A(G), and let
D(G) be the diagonal matrix of the degrees of G. The signless
Laplacian Q(G) of G is defined as Q(G):=A(G)+D(G).
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(G) into Q(G) in this paper it
is suggested to study the convex linear combinations Aα​(G) of A(G) and D(G) defined by Aα​(G):=αD(G)+(1−α)A(G),   0≤α≤1. This study sheds new light
on A(G) and Q(G), and yields some surprises, in
particular, a novel spectral Tur\'{a}n theorem. A number of challenging open
problems are discussed.Comment: 26 page