Let LDLt be the triangular factorization of an unreduced symmetric tridiagonal matrix T − τI . Small relative changes in the nontrivial entries of L and D may be represented by diagonal scaling matrices D1 and D2; LDL t −→ D2LD1DD1LD2. The effect of D2 on the eigenvalues λi − τ is benign. In this paper we study the inner perturbations induced by D1. Suitable condition numbers govern the relativ...

the center (periphery) of a graph is the set of vertices with minimum (maximum)eccentricity. in this paper, the structure of centers and peripheries of some classes ofcomposite graphs are determined. the relations between eccentricity, radius and diameterof such composite graphs are also investigated. as an application we determinethe center and periphery of some chemical graphs such as nanotor...

in this paper, a fundamentally new method, based on the denition, is introduced for numerical computation of eigenvalues, generalized eigenvalues and quadratic eigenvalues of matrices. some examples are provided to show the accuracy and reliability of the proposed method. it is shown that the proposed method gives other sequences than that of existing methods but they still are convergent to t...

The second Zagreb coindex is a well-known graph invariant defined as the total degree product of all non-adjacent vertex pairs in a graph. The second Zagreb eccentricity coindex is defined analogously to the second Zagreb coindex by replacing the vertex degrees with the vertex eccentricities. In this paper, we present exact expressions or sharp lower bounds for the second Zagreb eccentricity co...

Following the semiclassical formalism of Strutinsky et al. [11], we have obtained the complete eigenvalue spectrum for a particle enclosed in an infinitely high spheroidal cavity. Our spheroidal trace formula also reproduces the results of a spherical billiard in the limit η → 1.0. Inclusion of repetition of each family of the orbits with reference to the largest one significantly improves the ...

Let $L(G)$ be the Laplacian matrix of a digraph $G$ and $S_k(G)$ sum $k$ largest absolute values eigenvalues $G$. $C_n^+$ with $n+1$ vertices obtained from directed cycle $C_n$ by attaching pendant arc whose tail is on $C_n$. A $\mathbb{C}_n^+$-free if it contains no $C_{\ell}^+$ as subdigraph for any $2\leq \ell \leq n-1$. In this paper, we present lower bounds $S_n(G)$ digraphs order $n$. We ...

We consider generalizations of the k-source sum of vertex eccentricity problem (k-SVET) and the k-source sum of source eccentricity problem (k-SSET) [1], which we call SDET and SSET, respectively, and provide efficient algorithms for their solution. The SDET (SSET, resp.) problem is defined as follows: given a weighted graph G and sets S of source nodes and D of destination nodes, which are sub...

Let Xn be n×N containing i.i.d. complex entries and unit variance (sum of variances of real and imaginary parts equals 1), σ > 0 constant, and Rn an n×N random matrix independent of Xn. Assume, almost surely, as n →∞, the empirical distribution function (e.d.f.) of the eigenvalues of 1 N RnR ∗ n converges in distribution to a nonrandom probability distribution function (p.d.f.), and the ratio n...

In this paper we study the Betti numbers of a type of simplicial complex known as a chessboard complex. We obtain a formula for their Betti numbers as a sum of terms involving partitions. This formula allows us to determine which is the first nonvanishing Betti number (aside from the 0-th Betti number). We can therefore settle certain cases of a conjecture of Björner, Lovász, Vrećica, and Z̆ival...

We prove trace identities for commutators of operators, which are used to derive sum rules and sharp universal bounds for the eigenvalues of periodic Schrödinger operators and Schrödinger operators on immersed manifolds. In particular, we prove bounds on the eigenvalue λN+1 in terms of the lower spectrum, bounds on ratios of means of eigenvalues, and universal monotonicity properties of eigenva...