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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 A α ( G ) for the matrix A α ( G ) = α D ( G ) + ( 1 − α ) A ( G ) . Let α 0 ( G ) be the smallest α for which A α ( G ) is positive semidefinite. It is known that α 0 ( G ) ≤ 1 / 2 . The main results of this paper are: (1) if G is d -regular then α 0 = − λ min ( A ( G ) ) d − λ min ( A ( G ) ) , where λ min ( A ( G ) ) is the smallest eigenvalue of A ( G ) ; (2) G contains a bipartite component if and only if α 0 ( G ) = 1 / 2 ; (3) if G is r -colorable, then α 0 ( G ) ≥ 1 / r . [ABSTRACT FROM AUTHOR]

For a fixed real number r we give several necessary and/or sufficient conditions for a graph to have the second largest eigenvalue of the adjacency matrix, or signless Laplacian matrix, less then or equal to r . [ABSTRACT FROM AUTHOR]

Abstract: Consider a graph on vertices with adjacency matrix and degree sequence . A universal adjacency matrix of is any matrix in Span with a nonzero coefficient for , where and and are the identity and all-ones matrix, respectively. Thus a universal adjacency matrix is a common generalization of the adjacency, the Laplacian, the signless Laplacian and the Seidel matrix. We investigate graphs for which some universal adjacency matrix has just two eigenvalues. The regular ones are strongly regular, complete or empty, but several other interesting classes occur. [Copyright &y& Elsevier]