Merging the A-and Q-spectral theories

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). Cvetkovic called the study of the adjacency matrix the A-spectral theory, and the study of the signless Laplacian{the Q-spectral theory. To track the gradual change of A (G) into Q (G), in this paper it is suggested to study the convex linear combinations A_ (G) of A (G) and D (G) defined by A? (G): =? D (G) + (1 -? ) A (G), 0? ? ? 1. This study sheds new light on A (G) and Q (G), and yields, in particular, a novel spectral Tur? n theorem. A number of open problems are discussed.