scholarly journals Multipartite Separability of Laplacian Matrices of Graphs

10.37236/150 ◽  
2009 ◽  
Vol 16 (1) ◽  
Author(s):  
Chai Wah Wu
Keyword(s):  
Quantum Mechanics ◽  
Physical Properties ◽  
Laplacian Matrix ◽  
Bipartite Graphs ◽  
Density Matrices ◽  
Laplacian Matrices ◽  
Complete Bipartite Graphs ◽  
Vertex Degrees ◽  
Open Question ◽  
Product Of Graphs

Recently, Braunstein et al. introduced normalized Laplacian matrices of graphs as density matrices in quantum mechanics and studied the relationships between quantum physical properties and graph theoretical properties of the underlying graphs. We provide further results on the multipartite separability of Laplacian matrices of graphs. In particular, we identify complete bipartite graphs whose normalized Laplacian matrix is multipartite entangled under any vertex labeling. Furthermore, we give conditions on the vertex degrees such that there is a vertex labeling under which the normalized Laplacian matrix is entangled. These results address an open question raised in Braunstein et al. Finally, we show that the Laplacian matrix of any product of graphs (strong, Cartesian, tensor, lexicographical, etc.) is multipartite separable, extending analogous results for bipartite and tripartite separability.

scholarly journals Laplacian integral graphs with a given degree sequence constraint

2021 ◽  
Vol 40 (6) ◽  
pp. 1431-1448
Author(s):  
Ansderson Fernandes Novanta ◽  
Carla Silva Oliveira ◽  
Leonardo de Lima
Keyword(s):  
Adjacency Matrix ◽  
Degree Sequence ◽  
Laplacian Matrix ◽  
Bipartite Graphs ◽  
Diagonal Matrix ◽  
Connected Graphs ◽  
Sequence Constraint ◽  
The Matrix ◽  
Vertex Degrees ◽  
Integral Graphs

Let G be a graph on n vertices. The Laplacian matrix of G, denoted by L(G), is defined as L(G) = D(G) −A(G), where A(G) is the adjacency matrix of G and D(G) is the diagonal matrix of the vertex degrees of G. A graph G is said to be L-integral if all eigenvalues of the matrix L(G) are integers. In this paper, we characterize all Lintegral non-bipartite graphs among all connected graphs with at most two vertices of degree larger than or equal to three.

Laplacian integral graphs with a given degreee sequence constraint

2021 ◽  
Author(s):  
Anderson Fernandes Novanta ◽  
Carla Silva Oliveira ◽  
Leonardo Silva de Lima
Keyword(s):  
Adjacency Matrix ◽  
Laplacian Matrix ◽  
Bipartite Graphs ◽  
Diagonal Matrix ◽  
Connected Graphs ◽  
Sequence Constraint ◽  
The Matrix ◽  
Vertex Degrees ◽  
Integral Graphs

Let G be a graph on n vertices. The Laplacian matrix of G, denoted by L(G), is defined as L(G) = D(G) − A(G), where A(G) is the adjacency matrix of G and D(G) is the diagonal matrix of the vertex degrees of G. A graph G is said to be L-integral is all eigenvalues of the matrix L(G) are integers. In this paper, we characterize all L-integral non-bipartite graphs among all connected graphs with at most two vertices of degree larger than or equal to three.

Conjugate Laplacian matrices of a graph

2018 ◽  
Vol 10 (06) ◽  
pp. 1850082
Author(s):  
Somnath Paul
Keyword(s):  
Cartesian Product ◽  
Laplacian Matrix ◽  
Simple Graph ◽  
Discrete Mathematics ◽  
Laplacian Matrices ◽  
Cartesian Product Of Graphs ◽  
Laplacian Spectra ◽  
The Matrix ◽  
Matrix Formula ◽  
Product Of Graphs

Let [Formula: see text] be a simple graph of order [Formula: see text] Let [Formula: see text] and [Formula: see text] where [Formula: see text] and [Formula: see text] are two nonzero integers and [Formula: see text] is a positive integer such that [Formula: see text] is not a perfect square. In [M. Lepovi[Formula: see text], On conjugate adjacency matrices of a graph, Discrete Mathematics 307 (2007) 730–738], the author defined the matrix [Formula: see text] to be the conjugate adjacency matrix of [Formula: see text] if [Formula: see text] for any two adjacent vertices [Formula: see text] and [Formula: see text] for any two nonadjacent vertices [Formula: see text] and [Formula: see text] and [Formula: see text] if [Formula: see text] In this paper, we define conjugate Laplacian matrix of graphs and describe various properties of its eigenvalues and eigenspaces. We also discuss the conjugate Laplacian spectra for union, join and Cartesian product of graphs.

scholarly journals Permutational Powers of a Graph

10.37236/8337 ◽  
2019 ◽  
Vol 26 (4) ◽  
Author(s):  
Matteo Cavaleri ◽  
Daniele D'Angeli ◽  
Alfredo Donno
Keyword(s):  
Adjacency Matrix ◽  
Sufficient Conditions ◽  
Bipartite Graphs ◽  
Necessary And Sufficient Conditions ◽  
Permutation Matrix ◽  
Complete Bipartite Graphs ◽  
Necessary And Sufficient ◽  
Product Of Graphs

This paper introduces a new graph construction, the permutational power of a graph, whose adjacency matrix is obtained by the composition of a permutation matrix with the adjacency matrix of the graph. It is shown that this construction recovers the classical zig-zag product of graphs when the permutation is an involution, and it is in fact more general. We start by discussing necessary and sufficient conditions on the permutation and on the adjacency matrix of a graph to guarantee their composition to represent an adjacency matrix of a graph, then we focus our attention on the cases in which the permutational power does not reduce to a zig-zag product. We show that the cases of interest are those in which the adjacency matrix is singular. This leads us to frame our problem in the context of equitable partitions, obtained by identifying vertices having the same neighborhood. The families of cyclic and complete bipartite graphs are treated in details.

scholarly journals The Tripartite Separability of Density Matrices of Graphs

10.37236/958 ◽  
2007 ◽  
Vol 14 (1) ◽  
Author(s):  
Zhen Wang ◽  
Zhixi Wang
Keyword(s):  
Density Matrix ◽  
Laplacian Matrix ◽  
Point Graph ◽  
Density Matrices ◽  
Degree Condition ◽  
Laplacian Matrices ◽  
Combinatorial Laplacian ◽  
Nearest Point

The density matrix of a graph is the combinatorial laplacian matrix of a graph normalized to have unit trace. In this paper we generalize the entanglement properties of mixed density matrices from combinatorial laplacian matrices of graphs discussed in Braunstein et al. [Annals of Combinatorics, 10 (2006) 291] to tripartite states. Then we prove that the degree condition defined in Braunstein et al. [Phys. Rev. A, 73 (2006) 012320] is sufficient and necessary for the tripartite separability of the density matrix of a nearest point graph.

Assamese vowels and vowel harmony

2020 ◽  
Vol 6 (2) ◽  
pp. 151-183
Author(s):  
Diana B. Archangeli ◽  
Jonathan Yip
Keyword(s):  
Physical Properties ◽  
Principal Components Analysis ◽  
Principal Components ◽  
Vowel Harmony ◽  
Language Community ◽  
Acoustic Data ◽  
Apparent Loss ◽  
Root Position ◽  
Components Analysis ◽  
Open Question

AbstractBased on impressionistic and acoustic data, Assamese is described as having a phonological tongue root harmony system, with blocking by certain phonological configurations and over-application in certain morphological contexts. This study explores physical properties of the patterns using ultrasonic imaging to determine whether the impressionistic descriptions match what speakers actually do. Principal components analysis (PCA) determines that most participants produce a contrast in tongue root position in the appropriate contexts, though there is less of an impact on tongue root with greater distance from the triggering vowel. Analysis uses the root mean squared distance (RMSD) calculation to determine whether both blocking and over-application take effect. The blocking results conform to the impressionistic descriptions. With over-application, [e] and [o] are expected; while some speakers clearly produce these vowels, others articulate a vowel that is indeterminant between the expected [e]/[o] and an unexpected [ɛ]/[ɔ]. No speaker consistently showed the expected tongue root position in all contexts, and some speakers appeared to have lost the contrast entirely, yet all are considered to be speakers of the same dialect of Assamese. Whether this (apparent) loss is a consequence of crude research methodologies or accurately reflects what is happening within the language community remains an open question.

scholarly journals On acyclic edge-coloring of complete bipartite graphs

2017 ◽  
Vol 340 (3) ◽  
pp. 481-493
Author(s):  
Ayineedi Venkateswarlu ◽  
Santanu Sarkar ◽  
Sai Mali Ananthanarayanan
Keyword(s):  
Edge Coloring ◽  
Bipartite Graphs ◽  
Acyclic Edge Coloring ◽  
Complete Bipartite Graphs

On the error of approximations in quantum mechanics. I. General theory

1965 ◽  
Vol 257 (1081) ◽  
pp. 309-326 ◽  
Keyword(s):  
Quantum Mechanics ◽  
General Theory ◽  
Computer Applications ◽  
Quantum Mechanical ◽  
Quantum Mechanical Calculations ◽  
Density Matrices ◽  
Single Error

General formulas for estimating the errors in quantum-mechanical calculations are given in the formalism of density matrices. Some properties of the traces of matrices are used to simplify the estimating and to indicate a way of obtaining a better approximation. It is shown that the simultaneous correction of all the equations to be fulfilled leads in most cases to a faster convergence than the exact fulfilment of some of the equations and approximating stepwise to some of the others. The corrective formulas contain only direct operations of the matrices occurring and so they are advantageous in computer applications. In the last section a ‘subjective error’ definition is given and by taking into account the weight of the errors of the several equations a faster convergence and a single error quantity is obtained. Some special applications of the method will be published later.

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