The Laplace-Adomian Decomposition Method (LADM) and Homotopy Perturbation Method (HPM) are both utilized in this research in order to obtain an approximate analytical solution to the nonlinear Schrödinger equation with harmonic oscillator. Accordingly, nonlinear Schrödinger equation in both one and two dimensions is provided to illustrate the effects of harmonic oscillator on the behavior of the wave function. The available literature does not provide an exact solution to the problem presented in this paper. Nevertheless, approximate analytical solutions are provided in this paper using LADM and HPM methods, in addition to comparing and analyzing both solutions.
AbstractGiven the adjacency matrix A of a digraph, the eigenvalues of the matrix AAT constitute the so-called non-negative spectrum of this digraph. We investigate the relation between the structure of digraphs and their non-negative spectra and associated eigenvectors. In particular, it turns out that the non-negative spectrum of a digraph can be derived from the traditional (adjacency) spectrum of certain undirected bipartite graphs.
AbstractGiven any digraph D, its non-negative spectrum (or N-spectrum, shortly) consists of the eigenvalues of the matrix AAT, where A is the adjacency matrix of D. In this study, we relate the classical spectrum of undirected graphs to the N-spectrum of their oriented counterparts, permitting us to derive spectral bounds. Moreover, we study the spectral effects caused by certain modifications of a given digraph.
A free-form Sudoku puzzle is a square arrangement of m ×m cells such that the cells are partitioned into m subsets (called blocks) of equal cardinality. The goal of the puzzle is to place integers 1, . . , m in the cells such that the numbers in every row, column and block are distinct. Represent each cell by a vertex and add edges between two vertices exactly when the corresponding cells, according to the rules, must contain different numbers. This yields the associated free-form Sudoku graph. This article studies the eigenvalues of free-form Sudoku graphs, most notably integrality. Further, we analyze the evolution of eigenvalues and eigenspaces of such graphs when the associated puzzle is subjected to a ‘blow up’ operation, which scales the cell grid including its block partition.
The independent number of a graph G is the cardinality of the maximum independent set of G, denoted by α ( G ) . The independent dominating number is the cardinality of the smallest independent set that dominates all vertices of G. In this paper, we introduce a new class of graphs called semi-square stable for which α ( G 2 ) = i ( G ) . We give a necessary and sufficient condition for a graph to be semi-square stable, and we study when interval graphs are semi-square stable.
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