Graphs Of Matrices Research Articles

A note on automorphisms of the zero-divisor graph of upper triangular matrices

Let Fq be a finite field with q elements, n(≥3) a positive integer, T(n,q) the set of all n×n upper triangular matrices over Fq. In [13], the zero-divisor graph of T(n,q), written as T, is defined to be a graph with all nonzero zero-divisors in T(n,q) as vertices, and there is a directed edge from a vertex X to a vertex Y if and only if XY=0. The subgraph of T induced by all rank one matrices in T(n,q) is denoted by R. Wong et al. (2014) in [13] determined the automorphisms of R and left the automorphisms of T unsolved. In this note, we solve this problem.

Undirected graphs of Hermitian matrices that admit only two distinct eigenvalues

We consider the problem of determining those undirected n-vertex graphs with a corresponding Hermitian matrix that admits only two distinct eigenvalues, with multiplicities k and n−k. After giving some general algebraic characterizations of these dual multiplicity graphs, we then prove two major graph theoretic necessary conditions on such graphs. Construction techniques are then developed, and these lead to a characterization of dual multiplicity graphs for which the lesser multiplicity is two.

Chaotic Behaviors of Symbolic Dynamics about Rule 58 in Cellular Automata

The complex dynamical behaviors of rule 58 in cellular automata are investigated from the viewpoint of symbolic dynamics. The rule is Bernoulliστ-shift rule, which is members of Wolfram’s class II, and it was said to be simple as periodic before. It is worthwhile to study dynamical behaviors of rule 58 and whether it possesses chaotic attractors or not. It is shown that there exist two Bernoulli-measure attractors of rule 58. The dynamical properties of topological entropy and topological mixing of rule 58 are exploited on these two subsystems. According to corresponding strongly connected graph of transition matrices of determinative block systems, we divide determinative block systems into two subsets. In addition, it is shown that rule 58 possesses rich and complicated dynamical behaviors in the space of bi-infinite sequences. Furthermore, we prove that four rules of global equivalence classε43of CA are topologically conjugate. We use diagrams to explain the attractors of rule 58, where characteristic function is used to describe that some points fall into Bernoulli-shift map after several times iterations, and we find that these attractors are not global attractors. The Lameray diagram is used to show clearly the iterative process of an attractor.

Codes related to line graphs of triangular graphs and permutation decoding

For any prime p, we consider p-ary linear codes obtained from the row span of incidence matrices of line graphs of triangular graphs and adjacency matrices of their line graphs. We determine parameters of the codes, their automorphism groups and exhibit permutation decoding sets (PD-sets) for partial permutation decoding.

SIA Matrices and Non-Negative Subdivision

We reprove and extend results for uniform convergence of non-negative subdivision using results from infinite products of stochastic matrices as they appear in the study of non-homogeneous finite Markov processes. Geometric convergence of such products to rank one matrices, based on the notions of SIA matrices and of the so-called ergodic coefficient, is discussed. We also point to the properties of the directed graphs of such matrices. In this way the existing convergence results for non-negative subdivision are put into the context of such processes.

Diameters of commuting graphs of matrices over semirings

We calculate the diameters of commuting graphs of matrices over the binary Boolean semiring, the tropical semiring and an arbitrary nonentire commutative semiring. We also find the lower bound for the diameter of the commuting graph of the semigroup of matrices over an arbitrary commutative entire antinegative semiring.

Sharp bounds for sums associated to graphs of matrices

We provide a simple algorithm for finding the optimal upper bound for sums of products of matrix entries of the form S π ( N ) : = ∑ j 1 , … , j 2 m = 1 ker j ⩾ π N t j 1 j 2 ( 1 ) t j 3 j 4 ( 2 ) ⋯ t j 2 m − 1 j 2 m ( m ) where some of the summation indices are constrained to be equal. The upper bound is easily obtained from a graph G π associated to the constraints, π, in the sum.

Domination in Zero-divisor Graph of Matrices

Domination in Zero-divisor Graph of Matrices

Commuting graphs of matrices over semirings

We calculate diameters and girths of commuting graphs of the set of all nilpotent matrices over a semiring, the group of all invertible matrices over a semiring, and the full matrix semiring.

Noncommuting graphs of matrices over semirings

We study diameters and girths of noncommuting graphs of semirings. For a noncommutative semiring that is either multiplicatively or additively cancellative, we find the diameter and the girth of its noncommuting graph and prove that it is Hamiltonian. Moreover, we find diameters and girths of noncommuting graphs of all nilpotent matrices over a semiring, all invertible matrices over a semiring, all noninvertible matrices over a semiring, and the full matrix semiring. In nearly all cases we prove that diameters are less than or equal to 2 and girths are less than or equal to 3, except in the case of 2 × 2 nilpotent matrices.

Dual‐grid‐based tree/cotree decomposition of higher‐order interpolatory H(∇∧, Ω) basis

AbstractThis work extends the zeroth‐order tree/cotree (TC) decomposition method into higher order (HO) interpolatory elements and develops the constraints operator required for the elimination of spurious solutions for general HO spectral basis. Earlier methods explicitly enforce the divergence condition that requires a mixed finite element (FE) formulation with both H1 and H(∇∧) expansions and involves repeated solutions of the Poisson equation. A recent approach, which avoids the mixed formulation and the Poisson problem, uses TC decomposition of edge DoF over the primal graph and construction of integration and gradient matrices. The approach is easily applied to HO hierarchical elements but becomes quite complex for HO spectral elements. In the presence of internal DoF, it is difficult to utilize the primal graph for an explicit decomposition of the spectral DoF. In contrast, this work utilizes the dual grid, resulting in an explicit decomposition of DoF and construction of constraint equations from a fixed element matrix. Thus, mixed formulation and the Poisson problems are avoided while eliminating the need for evaluation of integration and gradient matrices. The proposed constraints matrix is element‐geometry independent and possesses an explicit sparsity formulation reducing the need for dynamic memory allocation. Numerical examples are included for verification. Copyright © 2010 John Wiley & Sons, Ltd.

On reduction of block matrices in a Hilbert space

We study the problem of the reduction of self-adjoint block matrices B = (B ij ) with given graph by a group of unitary block diagonal matrices. Under the condition that the matrices B 2 and B 4 are orthoscalar, we describe the graphs of block matrices for which this problem is a problem of *-finite, *-tame, or *-wild representation type.

A superclass of Edge-Path-Tree graphs with few cliques

Edge-Path-Tree (EPT) graphs are intersection graphs of EPT matrices that is matrices whose columns are incidence vectors of edge-sets of paths in a given tree. EPT graphs have polynomially many cliques [M.C. Golumbic, R.E. Jamison, The edge intersection graphs of paths in a tree, Journal of Combinational Theory Series B 38 (1985) 8–22; C.L. Monma, V.K. Wey, Intersection graphs of paths in a tree, Journal of Combinational Theory Series B 41 (1986) 141–181]. Therefore, the problem of finding a clique of maximum weight in these graphs is solvable in strongly polynomial time. We extend this result to a proper superclass of EPT graphs.

Conditions for weak ergodicity of inhomogeneous Markov chains

A classical result of Wolfowitz states that an inhomogeneous Markov chain is weakly ergodic if the transition matrices are drawn from a finite set of indecomposable and aperiodic matrices and the products of transition matrices are also indecomposable and aperiodic. Since products of indecomposable and aperiodic matrices can be decomposable, any finite set of indecomposable and aperiodic transition matrices does not guarantee weak ergodicity. We present conditions for weak ergodicity which are simpler to verify and are related to properties of the graph of the transition matrices.

Relation Between Parity-Check Matrixes and Cycles of Associated Tanner Graphs

This correspondence presents an approach to the representation of cycles of Tanner graphs on associated parity-check matrices. Several equivalent conditions for the girth of a Tanner graph to be 2k are proposed. An algorithm to determine girth of associated Tanner graphs of parity-check matrices is proposed, also with an algorithm to count shortest cycles.

Weighted complexities of graph products and bundles

The complexity κ(G) of a graph G is the number of spanning trees in G. In spite of its importance, most known methods for computing κ(G) commonly have computational difficulties since they require to compute determinants or eigenvalues of matrices of the size of the order of a graph. In particular, they are not feasible for large graphs. However, many of them can be represented by some graph operations. A graph bundle is a notion containing a cartesian product of graphs and a (regular or irregular) graph covering. For a regular graph covering, H. Mizuno and I. Sato [Zeta functions for images of graph coverings by some operations, Interdiscip. Inform. Sci. 7 (2001) 53–60] computed its complexity. We extend their work to a graph bundle by deriving a factorized formula for the complexity: If a graph bundle has a regular fibre, its complexity can be factorized into the complexity of the base graph and determinants of smaller-size matrices. For the complexities of the cartesian products of graphs, several computing formulae are already known. However, they also used somewhat complicated calculations of determinants, eigenvalues or trigonometric equations. We reduce such complication for the known cases of the ladder, the Möbius ladder and the prism, by simply deriving the factorized formulae for their complexities. New concrete formulae for the complexities of the product Pn×Km of the path Pn and the complete graph Km and those ofKm-bundles over the cycle Cn are also derived as generalizations of the prism and the Möbius ladder.

Determination of sulphur in liquids obtained by thermal cracking of waste polymers and commercial fuels with different analytical methods

Low sulphur concentration in hydrocarbon products as fuels or lubricants is an important requirement for the high quality standards of refineries. A non-polarised energy dispersive X-ray fluorescence spectroscopy (EDXRFS) and sample combustion technique (ASTM D6428-99) was compared. A new application of energy dispersive X-ray spectrometry as analytical method for the determination of sulphur in fuels and fuel-like fractions was investigated. Low sulphur containing fuels and hydrocarbon mixtures obtained by thermal cracking of waste polymers were measured and the influence of C/H ratio on accuracy was studied. The concentration of sulphur in samples was measured with calibration graphs of different hydrocarbon matrices (commercial gasoline, diesel oil and white oil were used). Good correlation was observed between the different methods, but the correlation was depending on the characteristics of the matrices. Detection limits of 1.0 ppm, 1.1 ppm and 0.9 ppm were obtained for S in gasoline, diesel oil and white oil, respectively.

Structural properties of positive periodic discrete-time linear systems: canonical forms

The reachability and controllability properties of positive periodic discrete-time linear systems are studied. Using the directed graph of the state matrices and the concepts of colored vertices and colored union of directed graphs, a characterization of these properties is established and hence, canonical forms are deduced.

Distances between the graphs of matrices

The graph of a complex m-by- n matrix A is the range of [ I A T ] T in C m+ n . We find explicit formulae for the distance between the graphs of two m-by- n matrices with respect to two classes of metrics on the set of subspaces of C m+ n . The singular values of the contraction g(A, B)  (I+A∗ A) −1 2 (I+A∗ B)(I+B∗ B) −1 2 play a key role. We study transformations that preserve various distances between graphs, and we give some inequalities between different distance functions.

Parsons graphs of matrices on [formula omitted

Parsons graphs of matrices on [formula omitted