Sinc-Self-consistent method to solve a class of nonlinear eigenvalue differential equation

In this paper, we combine the sinc and self-consistent methods to solve a class of non-linear eigenvalue differential equations. Some properties of the self-consistent and sinc methods required for our subsequent development are given and employed. Numerical examples are included to demonstrate the validity and applicability of the introduced technique and a comparison is made with the existing results. The method is easy to implement and yields accurate results. We show that the sinc-self-consistent method can solve the equations on an infinite domain and produces the smallest eigenvalue with the most accuracy

On the Spectrum of Laplacian Matrix

Let G be a simple graph of order n . The matrix ℒ G = D G − A G is called the Laplacian matrix of G , where D G and A G denote the diagonal matrix of vertex degrees and the adjacency matrix of G , respectively. Let l 1 G , l n − 1 G be the largest eigenvalue, the second smallest eigenvalue of ℒ G respectively, and λ 1 G be the largest eigenvalue of A G . In this paper, we will present sharp upper and lower bounds for l 1 G and l n − 1 G . Moreover, we investigate the relation between l 1 G and λ 1 G .

Neumaier graphs with few eigenvalues

A Neumaier graph is a non-complete edge-regular graph containing a regular clique. In this paper we give some sufficient and necessary conditions for a Neumaier graph to be strongly regular. Further we show that there does not exist Neumaier graphs with exactly four distinct eigenvalues. We also determine the Neumaier graphs with smallest eigenvalue $$-2$$ - 2 .

On the smallest singular value in the class of unit lower triangular matrices with entries in [−a, a]

Given a real number a ≥ 1, let Kn (a) be the set of all n × n unit lower triangular matrices with each element in the interval [−a, a]. Denoting by λn (·) the smallest eigenvalue of a given matrix, let cn (a) = min {λ n (YYT ) : Y ∈ Kn (a)}. Then c n ( a ) \sqrt {{c_n}\left( a \right)} is the smallest singular value in Kn (a). We find all minimizing matrices. Moreover, we study the asymptotic behavior of cn (a) as n → ∞. Finally, replacing [−a, a] with [a, b], a ≤ 0 < b, we present an open question: Can our results be generalized in this extension?