6 N ov 2 01 6 A note on the positive semidefinitness of A α ( G ) Vladimir
نویسنده
چکیده
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. AMS classification: 05C50, 15A48
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تاریخ انتشار 2016