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Let G be a simple graph with the adjacency matrix A (G) , and let τ (G) denote the smallest positive eigenvalue of A (G). Let G n be the class of all connected bipartite graphs on n = 2 k vertices with a unique perfect matching. In this article, we characterize the graphs G in G n such that τ (G) does not exceed 1 2. Using the above characterization, we obtain the unique graphs in G n with the maximum and the second maximum τ , respectively. Further, we prove that the largest and the second largest limit points of the smallest positive eigenvalues of bipartite graphs with a unique perfect matching are 1 2 and the reciprocal of α 3 1 2 + α 3 − 1 2 , respectively, where α 3 is the largest root of x 3 − x − 1. [ABSTRACT FROM AUTHOR]

A T -gain graph, Φ = (G , φ) , is a graph in which the function φ assigns a unit complex number to each orientation of an edge of G , and its inverse is assigned to the opposite orientation. The associated adjacency matrix A (Φ) is defined canonically. The energy E (Φ) of a T -gain graph Φ is the sum of the absolute values of all eigenvalues of A (Φ). We study the notion of energy of a vertex of a T -gain graph, and establish bounds for it. For any T -gain graph Φ, we prove that 2 τ (G) − 2 c (G) ≤ E (Φ) ≤ 2 τ (G) Δ (G) , where τ (G) , c (G) and Δ (G) are the vertex cover number, the number of odd cycles and the largest vertex degree of G , respectively. Furthermore, using the properties of vertex energy, we characterize the class of T -gain graphs for which E (Φ) = 2 τ (G) − 2 c (G) holds. Also, we characterize the T -gain graphs for which E (Φ) = 2 τ (G) Δ (G) holds. This characterization solves a general version of an open problem. In addition, we establish bounds for the energy in terms of the spectral radius of the associated adjacency matrix. [ABSTRACT FROM AUTHOR]

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]

Let k , p , q be positive integers with k < p < q + 1 . We prove that the maximum spectral radius of a simple bipartite graph obtained from the complete bipartite graph K p , q of bipartition orders p and q by deleting k edges is attained when the deleted edges are all incident on a common vertex which is located in the partite set of order q . Our method is based on new sharp upper bounds on the spectral radius of bipartite graphs in terms of their degree sequences. [ABSTRACT FROM AUTHOR]