A note on the positive semidefiniteness of Aα(G)

Let G be a graph with adjacency matrix A(G) and let D(G) be the diagonal matrix of the degrees of G. For every real α∈[0,1], write Aα(G) for the matrixAα(G)=αD(G)+(1−α)A(G). Let α0(G) be the smallest α for which Aα(G) is positive semidefinite. It is known that α0(G)≤1/2. The main results of this paper are: