Merging the A- and Q-spectral theories

Abstract

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). Cvetković 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 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, in particular, a novel spectral Turán theorem. A number of open problems are discussed.