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 matrix Aα(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: (1) if G is d-regular then α0=−λmin(A(G))d−λmin(A(G)), where λmin(A(G)) is the smallest eigenvalue of A(G);(2) G contains a bipartite component if and only if α0(G)=1/2;(3) if G is r-colorable, then α0(G)≥1/r.

Publication Title

Linear Algebra and Its Applications