Let k be a positive integer and let G be a graph of order n ≥ k. It is proved that the sum of k largest eigenvalues of G is at most 1 2 ( √ k+1)n. This bound is shown to be best possible in the sense that for every k there exist graphs whose sum is 12 ( √ k + 12 )n− o(k−2/5)n. A generalization to arbitrary symmetric matrices is given.

We compare two different approaches to investigations of many-electron systems. The first one is the Hartree-Fock (HF) method and the second one is the Density Functional Theory (DFT). Overview of the main features and peculiar properties of HF method is presented. We discuss a way to realize the HF method within the Kohn-Sham (KS) theory and show that it is impossible without including a speci...

Let G be a graph with two non adjacent vertices and G′ the graph constructed from G by adding an edge between them. It is known that the trace of Q′ is 2 plus the trace of Q, where Q and Q′ are the signless Laplacian matrices of G and G′ respectively. So, the sum of the Q′-eigenvalues of G′ is the sum of the the Qeigenvalues of G plus two. It is said that Q-spectral integral variation occurs wh...

The distance energy of a simple connected graph G is defined as the sum absolute values its eigenvalues. In this paper, we mainly give positive answer to conjecture clique trees proposed by Lin et al. (2015) [16].

Using the Green's function associated with the one-dimensional Schroedinger equation it is possible to establish a hierarchy of sum rules involving the eigenvalues of confining potentials which have only a boundstate spectrum. For some potentials the sum rules could lead to divergences. It is shown that when this happens it is possible to examine the separate sum rules satisfied by the even and...

Our purpose in this paper is to study of the eigenvalues {?i(?)}i Dirichlet problem(??)s1u=?((??)s2u+?u)in?,u=0inRN??, where 02s1 and (??)s fractional Laplacian operator defined principle value sense. We first show existence a sequence eigenvalues, which approaches infinity. Secondly we provide Berezin–Li–Yau type lower bound for sum above problem. Furthermore, using self-contained ...

We derive simplified formulas for analyzing the moment stability of stochastic parametrically forced linear systems. This analysis extends the results in [3], where, assuming the stochastic excitation is small, the stability of such systems was computed using a weighted sum of the extended power spectral density over the eigenvalues of the unperturbed operator. In this paper, we show how to con...