38 research outputs found
On backward errors of structured polynomial eigenproblems solved by structure preserving linearizations
First, we derive explicit computable expressions of structured backward
errors of approximate eigenelements of structured matrix polynomials including
symmetric, skew-symmetric, Hermitian, skew-Hermitian, even and odd polynomials.
We also determine minimal structured perturbations for which approximate
eigenelements are exact eigenelements of the perturbed polynomials. Next, we
analyze the effect of structure preserving linearizations of structured matrix
polynomials on the structured backward errors of approximate eigenelements. We
identify structure preserving linearizations which have almost no adverse
effect on the structured backward errors of approximate eigenelements of the
polynomials. Finally, we analyze structured pseudospectra of a structured
matrix polynomial and establish a partial equality between unstructured and
structured pseudospectra.Comment: 27 pages, submitte
Structured mapping problems for linearly structured matrices
Given an appropriate class of structured matrices S; we characterize matrices
X and B for which there exists a matrix A \in S such that AX = B and determine
all matrices in S mapping X to B. We also determine all matrices in S mapping X
to B and having the smallest norm. We use these results to investigate
structured backward errors of approximate eigenpairs and approximate invariant
subspaces, and structured pseudospectra of structured matrices.Comment: 15 page
Structured inverse least-squares problem for structured matrices
Given a pair of matrices X and B and an appropriate class of structured
matrices S, we provide a complete solution of the structured inverse
least-squares problem . Indeed, we determine all
solutions of the structured inverse least squares problem as well as those
solutions which have the smallest norm. We show that there are infinitely many
smallest norm solutions of the least squares problem for the spectral norm
whereas the smallest norm solution is unique for the Frobenius norm.Comment: 10 pages. arXiv admin note: text overlap with arXiv:1309.252
Self-Coordinated Corona Graphs: a model for complex networks
Recently, real world networks having constant/shrinking diameter along with
power-law degree distribution are observed and investigated in literature.
Taking an inspiration from these findings, we propose a deterministic complex
network model, which we call Self-Coordinated Corona Graphs (SCCG), based on
the corona product of graphs. As it has also been established that self
coordination/organization of nodes gives rise to emergence of power law in
degree distributions of several real networks, the networks in the proposed
model are generated by the virtue of self coordination of nodes in corona
graphs. Alike real networks, the SCCG inherit motifs which act as the seed
graphs for the generation of SCCG. We also analytically prove that the power
law exponent of SCCG is approximately and the diameter of SCCG produced by
a class of motifs is constant. Finally, we compare different properties of the
proposed model with that of the BA and Pseudofractal scale-free models for
complex networks.Comment: 21 pages, 31 figure
Context dependent preferential attachment model for complex networks
In this paper, we propose a growing random complex network model, which we
call context dependent preferential attachment model (CDPAM), when the
preference of a new node to get attached to old nodes is determined by the
local and global property of the old nodes. We consider that local and global
properties of a node as the degree and relative average degree of the node
respectively. We prove that the degree distribution of complex networks
generated by CDPAM follow power law with exponent lies in the interval [2, 3]
and the expected diameter grows logarithmically with the size of new nodes
added in the initial small network. Numerical results show that the expected
diameter stabilizes when alike weights to the local and global properties are
assigned by the new nodes. Computing various measures including clustering
coefficient, assortativity, number of triangles, algebraic connectivity,
spectral radius, we show that the proposed model replicates properties of real
networks better than BA model for all these measures when alike weights are
given to local and global property. Finally, we observe that the BA model is a
limiting case of CDPAM when new nodes tend to give large weight to the local
property compared to the weight given to the global property during link
formation.Comment: 07 page
Laplacian matrices of weighted digraphs represented as quantum states
Representing graphs as quantum states is becoming an increasingly important
approach to study entanglement of mixed states, alternate to the standard
linear algebraic density matrix-based approach of study. In this paper, we
propose a general weighted directed graph framework for investigating
properties of a large class of quantum states which are defined by three types
of Laplacian matrices associated with such graphs. We generalize the standard
framework of defining density matrices from simple connected graphs to density
matrices using both combinatorial and signless Laplacian matrices associated
with weighted directed graphs with complex edge weights and with/without
self-loops. We also introduce a new notion of Laplacian matrix, which we call
signed Laplacian matrix associated with such graphs. We produce necessary
and/or sufficient conditions for such graphs to correspond to pure and mixed
quantum states. Using these criteria, we finally determine the graphs whose
corresponding density matrices represent entangled pure states which are well
known and important for quantum computation applications. We observe that all
these entangled pure states share a common combinatorial structure.Comment: 19 pages, Modified version of quant-ph/1205.2747, title and abstract
has been changed, One author has been adde
Backward errors and linearizations for palindromic matrix polynomials
We derive computable expressions of structured backward errors of approximate
eigenelements of *-palindromic and *-anti-palindromic matrix polynomials. We
also characterize minimal structured perturbations such that approximate
eigenelements are exact eigenelements of the perturbed polynomials. We detect
structure preserving linearizations which have almost no adverse effect on the
structured backward errors of approximate eigenelements of the *-palindromic
and *-anti-palindromic polynomials.Comment: 19 pages, submitte
Structural and Spectral properties of Corona Graphs
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 seed graph, Corona graphs are defined by taking
corona product of a seed graph iteratively. We show that the cumulative degree
distribution of Corona graphs decay exponentially when the seed graph is
regular and cumulative betweenness distribution follows power law when seed
graph is a clique. We determine explicit formulae of eigenvalues, Laplacian
eigenvalues and signless Laplacian eigenvalues of Corona graphs when the seed
graph is regular. Computable expressions of eigenvalues and signless Laplacian
eigenvalues of Corona graphs are also obtained when the seed graph is a star
graph.Comment: 24 pages, 12 figure
A graph theoretical approach to states and unitary operations
Building upon our previous work, on graphical representation of a quantum
state by signless Laplacian matrix, we pose the following question. If a local
unitary operation is applied to a quantum state, represented by a signless
Laplacian matrix, what would be the corresponding graph and how does one
implement local unitary transformations graphically? We answer this question by
developing the notion of local unitary equivalent graphs. We illustrate our
method by a few, well known, local unitary transformations implemented by
single-qubit Pauli and Hadamard gates. We also show how graph switching can be
used to implement the action of the CNOT gate, resulting in a graphical
description of Bell state generation.Comment: 20 pages, version very similar to the one published in quantum
information processin
Condition for zero and non-zero discord in graph Laplacian quantum states
This work is at the interface of graph theory and quantum mechanics. Quantum
correlations epitomize the usefulness of quantum mechanics. Quantum discord is
an interesting facet of bipartite quantum correlations. Earlier, it was shown
that every combinatorial graph corresponds to quantum states whose
characteristics are reflected in the structure of the underlined graph. A
number of combinatorial relations between quantum discord and simple graphs
were studied. To extend the scope of these studies, we need to generalize the
earlier concepts applicable to simple graphs to weighted graphs, corresponding
to a diverse class of quantum states. To this effect, we determine the class of
quantum states whose density matrix representation can be derived from graph
Laplacian matrices associated with a weighted directed graph and call them
graph Laplacian quantum states. We find the graph-theoretic conditions for zero
and non-zero quantum discord for these states. We apply these results on some
important pure two qubit states, as well as a number of mixed quantum states,
such as the Werner, Isotropic, and -states. We also consider graph Laplacian
states corresponding to simple graphs as a special case.Comment: 24 pages, this version is very similar to the one published in the
International Journal of Quantum Informatio