Product graphs have been gainfully used in literature to generate mathematical models of complex networks which inherit properties of real networks. Realizing the duplication phenomena imbibed in the definition of corona product of two graphs, we define corona graphs. Given a small simple connected graph which we call basic graph, corona graphs are defined by taking corona product of the basic ...

We study the relationship between the eigenvalues of separated self-adjoint boundary conditions and coupled self-adjoint conditions. Given an arbitrary real coupled boundary condition determined by a coupling matrix K we construct a one parameter family of separated conditions and show that all the eigenvalues for K and −K are extrema of the eigencurves of this family. This characterization mak...

The problem of finding an energy conserving norm for the solution of the hyperbolic system of partial differential equations du/dt = Adu/dx, subject to boundary conditions, is reduced to the problem of characterizing those matrices appearing in the boundary conditions which satisfy two specific matrix equations. Necessary and sufficient conditions on the coefficient matrix A and the matrices ap...

We explicitly determine the skew-symmetric eigenvectors and corresponding eigenvalues of the real symmetric Toeplitz matrices T = T (a, b, n) := (a+ b|j − k|)1≤j,k≤n of order n ≥ 3 where a, b ∈ R, b 6= 0. The matrix T is singular if and only if c := a b = −n−1 2 . In this case we also explicitly determine the symmetric eigenvectors and corresponding eigenvalues of T . If T is regular, we explic...

For selfadjoint matrices in an indefinite inner product, possible canonical forms are identified that arise when the matrix is subjected to a selfadjoint generic rank one perturbation. Genericity is understood in the sense of algebraic geometry. Special attention is paid to the perturbation behavior of the sign characteristic. Typically, under such a perturbation, for every given eigenvalue, th...

According to Kippenhahn’s classification, numerical ranges W (A) of unitarily irreducible 3 × 3 matrices A come in three possible shapes, an elliptical disk being one of them. The known criterion for the ellipticity of W (A) consists of several equations, involving the eigenvalues of A. It is shown herein that the set of 3× 3 matrices satisfying these conditions is nowhere dense, i.e., one of t...

So far, various components of image characteristics have been used for steganalysis, including the histogram characteristic function, adjacent colors distribution, and sample pair analysis. However, some certain steganography methods have been proposed that can thwart some analysis approaches through managing the embedding patterns. In this regard, the present paper is intended to introduce a n...

Currently, it has been claimed that certain Hermitian Hamiltonians have parity (P) and they are PT-invariant. We propose generalized definitions of time-reversal operator (T) and orthonormality such that all Hermitian Hamil-tonians are P, T, PT, and CPT invariant. The PT-norm and CPT-norm are indefinite and definite respectively. The energy-eigenstates are either E-type (e.g., even) or O-type (...

Let G be a graph of order n with signless Laplacian eigenvalues q1, . . . , qn and Laplacian eigenvalues μ1, . . . , μn. It is proved that for any real number α with 0 < α 6 1 or 2 6 α < 3, the inequality qα 1 + · · · + qα n > μ1 + · · · + μn holds, and for any real number β with 1 < β < 2, the inequality q 1 + · · ·+ q n 6 μβ1 + · · ·+ μ β n holds. In both inequalities, the equality is attaine...

The FEAST algorithm [5] is an algorithm aimed at solving generalized eigenvalue problems AX = BXΛ, where A,B are n×n matrices. In [4] we presented a short analysis of FEAST involving this eigenvalue problem. In the present work we focus on the real symmetric eigenvalue problem AX = XΛ, where A = A, XX = I and Λ is a diagonal matrix consisting of real eigenvalues. These eigenvalues are sought in...