Graph-theoretic Properties Research Articles

Divergence zero quaternionic vector fields and Hamming graphs

We give a possible extension of the definition of quaternionic power series, partial derivatives and vector fields in the case of two (and then several) non commutative (quaternionic) variables. In this setting we also investigate the problem of describing zero functions which are not null functions in the formal sense. A connection between an analytic condition and a graph theoretic property of a subgraph of a Hamming graph is shown, namely the condition that polynomial vector field has formal divergence zero is equivalent to connectedness of subgraphs of Hamming graphs H ( d , 2) . We prove that monomials in variables z and w are always linearly independent as functions only in bidegrees ( p , 0), ( p , 1), (0, q ), (1, q ) and (2, 2).

Construction of embedded fMRI resting-state functional connectivity networks using manifold learning

We construct embedded functional connectivity networks (FCN) from benchmark resting-state functional magnetic resonance imaging (rsfMRI) data acquired from patients with schizophrenia and healthy controls based on linear and nonlinear manifold learning algorithms, namely, Multidimensional Scaling, Isometric Feature Mapping, Diffusion Maps, Locally Linear Embedding and kernel PCA. Furthermore, based on key global graph-theoretic properties of the embedded FCN, we compare their classification potential using machine learning. We also assess the performance of two metrics that are widely used for the construction of FCN from fMRI, namely the Euclidean distance and the cross correlation metric. We show that diffusion maps with the cross correlation metric outperform the other combinations.

Connectivity of generating graphs of nilpotent groups

Let $G$ be $2$-generated group. The generating graph of $\Gamma(G)$ is the graph whose vertices are the elements of $G$ and where two vertices $g$ and $h$ are adjacent if $G=\langle g,h\rangle$. This graph encodes the combinatorial structure of the distribution of generating pairs across $G$. In this paper we study several natural graph theoretic properties related to the connectedness of $\Gamma(G)$ in the case where $G$ is a finite nilpotent group. For example, we prove that if $G$ is nilpotent, then the graph obtained from $\Gamma(G)$ by removing its isolated vertices is maximally connected and, if $|G| \geq 3$, also Hamiltonian. We pose several questions.

Perpendicular graph of modules

Let $R$ be a ring and $M$ be an $R$-module. Two modules $A$ and $B$ are called orthogonal, written $A\perp B$, if they do not have non-zero isomorphic submodules. We associate a graph $\Gamma_{\bot}(M)$ to $M$ with vertices $\mathcal{M}_{\perp}=\{(0)\neq A\leq M \mid \exists B\neq (0) \;\mbox{such that}\; A\perp B\}$, and for distinct $A,B\in \mathcal{M}_{\perp}$, the vertices $A$ and $B$ are adjacent if and only if $A\perp B$. The main object of this article is to study the interplay of module-theoretic properties of $M$ with graph-theoretic properties of $\Gamma_{\bot}(M)$. An algorithm is given to generate perpendicular graphs of $\mathbb{Z}_n$.

Morphophoric POVMs, generalised qplexes, and 2-designs

We study the class of quantum measurements with the property that the image of the set of quantum states under the measurement map transforming states into probability distributions is similar to this set and call such measurements morphophoric. This leads to the generalisation of the notion of a qplex, where SIC-POVMs are replaced by the elements of the much larger class of morphophoric POVMs, containing in particular 2-design (rank-1 and equal-trace) POVMs. The intrinsic geometry of a generalised qplex is the same as that of the set of quantum states, so we explore its external geometry, investigating, inter alia, the algebraic and geometric form of the inner (basis) and the outer (primal) polytopes between which the generalised qplex is sandwiched. In particular, we examine generalised qplexes generated by MUB-like 2-design POVMs utilising their graph-theoretical properties. Moreover, we show how to extend the primal equation of QBism designed for SIC-POVMs to the morphophoric case.

Mapping graph state orbits under local complementation

Graph states, and the entanglement they posses, are central to modern quantum computing and communications architectures. Local complementation – the graph operation that links all local-Clifford equivalent graph states – allows us to classify all stabiliser states by their entanglement. Here, we study the structure of the orbits generated by local complementation, mapping them up to 9 qubits and revealing a rich hidden structure. We provide programs to compute these orbits, along with our data for each of the 587 orbits up to 9 qubits and a means to visualise them. We find direct links between the connectivity of certain orbits with the entanglement properties of their component graph states. Furthermore, we observe the correlations between graph-theoretical orbit properties, such as diameter and colourability, with Schmidt measure and preparation complexity and suggest potential applications. It is well known that graph theory and quantum entanglement have strong interplay – our exploration deepens this relationship, providing new tools with which to probe the nature of entanglement.

The Importance of Anti-correlations in Graph Theory Based Classification of Autism Spectrum Disorder.

With the release of the multi-site Autism Brain Imaging Data Exchange, many researchers have applied machine learning methods to distinguish between healthy subjects and autistic individuals by using features extracted from resting state functional MRI data. An important part of applying machine learning to this problem is extracting these features. Specifically, whether to include negative correlations between brain region activities as relevant features and how best to define these features. For the second question, the graph theoretical properties of the brain network may provide a reasonable answer. In this study, we investigated the first issue by comparing three different approaches. These included using the positive correlation matrix (comprising only the positive values of the original correlation matrix), the absolute value of the correlation matrix, or the anticorrelation matrix (comprising only the negative correlation values) as the starting point for extracting relevant features using graph theory. We then trained a multi-layer perceptron in a leave-one-site out manner in which the data from a single site was left out as testing data and the model was trained on the data from the other sites. Our results show that on average, using graph features extracted from the anti-correlation matrix led to the highest accuracy and AUC scores. This suggests that anti-correlations should not simply be discarded as they may include useful information that would aid the classification task. We also show that adding the PCA transformation of the original correlation matrix to the feature space leads to an increase in accuracy.

K-Fibonacci Cubes: A Family of Subgraphs of Fibonacci Cubes

Hypercubes and Fibonacci cubes are classical models for interconnection networks with interesting graph theoretic properties. We consider [Formula: see text]-Fibonacci cubes, which we obtain as subgraphs of Fibonacci cubes by eliminating certain edges during the fundamental recursion phase of their construction. These graphs have the same number of vertices as Fibonacci cubes, but their edge sets are determined by a parameter [Formula: see text]. We obtain properties of [Formula: see text]-Fibonacci cubes including the number of edges, the average degree of a vertex, the degree sequence and the number of hypercubes they contain.

Synchronizing sequences for road colored digraphs

In this paper we consider the Černý conjecture from the point of view of colored digraphs corresponding to automata. We prove this conjecture for a new class of automata that is defined in terms of graph-theoretic properties related to crossing paths of different colors in colored digraphs. This yields not only a new class of automata for which the conjecture is verified, but also gives some novel insight into the structure and complexity of the whole problem.

Resilience and technological diversity in smart homes

This article introduces our abstract modeling strategy to represent the general features and topology of the kinds of integrated and technologically diverse networks that feature in IoT systems. We begin with smart home networks. We generate instances of our model and analyze their graph-theoretic properties with an emphasis on the resilience of critical services and connections to the Global Internet. In addition to considering the network connectivity graph of nodes and links in the model, we explain our technology interdependence graph techniques. Technology interdependence graphs allow us to illuminate critical interactions in multi-technology systems such as smart homes. Using relatively simple examples we show how our approach permits the exploration of the resilience properties of various instances of smart systems involving complex technological interdependency. We describe a practical way of approaching the graphs of systems with a wide variety of integrated technologies and we discuss properties such as connectedness and other metrics. This approach can serve as the basis for tackling the challenge of designing resilient IoT-based smart-cities from the point of view of network topologies. We also study smart home resilience through path redundancy and heterogeneity of network technologies with graph centrality metrics.

Graph-theoretic Simplification of Quantum Circuits with the ZX-calculus

We present a completely new approach to quantum circuit optimisation, based on the ZX-calculus. We first interpret quantum circuits as ZX-diagrams, which provide a flexible, lower-level language for describing quantum computations graphically. Then, using the rules of the ZX-calculus, we give a simplification strategy for ZX-diagrams based on the two graph transformations of local complementation and pivoting and show that the resulting reduced diagram can be transformed back into a quantum circuit. While little is known about extracting circuits from arbitrary ZX-diagrams, we show that the underlying graph of our simplified ZX-diagram always has a graph-theoretic property called generalised flow, which in turn yields a deterministic circuit extraction procedure. For Clifford circuits, this extraction procedure yields a new normal form that is both asymptotically optimal in size and gives a new, smaller upper bound on gate depth for nearest-neighbour architectures. For Clifford+T and more general circuits, our technique enables us to to `see around' gates that obstruct the Clifford structure and produce smaller circuits than naïve `cut-and-resynthesise' methods.

Some Remarks on the Annihilator Graph of a Commutative Ring

Let $R$ be a commutative ring with nonzero identity. The annihilator graph of $R$, denoted by $AG(R)$, is the (undirected) graph whose vertex set is the set of all nonzero zero-divisors of $R$ and two distinct vertices $x$ and $y$ are adjacent if and only if $\ann_R(xy)\neq {\rm ann}_R(x)\cup {\rm ann}_R(y)$. We investigate the interplay between ring-theoretic properties of $R$ and graph-theoretic properties of $AG(R)$. We study the relation between two graphs $\Gamma(R)$ and $AG(R)$, where $R$ is a non-reduced commutative ring. Also, we completely characterize the rings whose annihilator graphs are complete.

Min-max and max-min graph saturation parameters

Let P be a graph theoretic property concerning subsets of the vertex set V, which is either hereditary or super hereditary. We associate with this property P five graph theoretic parameters, namely a minimum parameter, a maximum parameter, a max-min parameter, a min-max parameter and a partition parameter. We initiate a study of these parameters for the properties of independence, domination and irredundance.

Complement of the generalized total graph of fields

Let R be a commutative ring and H be a multiplicative prime subset of R. The generalized total graph is the undirected simple graph with vertex set R and two distinct vertices x and y are adjacent if For a field F, is the only multiplicative prime subset of F and the corresponding generalized total graph is denoted by In this paper, we investigate several graph theoretical properties of where is the complement of the generalized total graph of F. In particular, we characterize all the fields for which is unicyclic, split, chordal, claw-free, perfect and pancyclic.

Nilpotent graphs of skew polynomial rings over non-commutative rings

Let $R$ be a ring and $alpha$ be a ring endomorphism of $R$‎. ‎The undirected nilpotent graph of $R$‎, ‎denoted by $Gamma_N(R)$‎, ‎is a graph with vertex set $Z_N(R)^*$‎, ‎and two distinct vertices $x$ and $y$ are connected by an edge if and only if $xy$ is nilpotent‎, ‎where $Z_N(R)={xin R;|; xy; rm{is; nilpotent,;for; some}; yin R^*}.$ In this article‎, ‎we investigate the interplay between the ring theoretical properties of a skew polynomial ring $R[x;alpha]$ and the graph-theoretical properties of its nilpotent graph $Gamma_N(R[x;alpha])$‎. ‎It is shown that if $R$ is a symmetric and $alpha$-compatible with exactly two minimal primes‎, ‎then $diam(Gamma_N(R[x,alpha]))=2$‎. ‎Also we prove that $Gamma_N(R)$ is a complete graph if and only if $R$ is isomorphic to $Z_2timesZ_2$‎.

THE TOTAL GRAPH OF ANNIHILATING ONE-SIDED IDEALS OF A RING

Let $R$ be an associative ring with $1\neq 0$ which is not a domain. Let $A(R)^*=\{I\subseteq R~|~I \text{ is a left or right ideal of } R \text{ and } \mathrm{l.ann}(I)\cup \mathrm{r.ann}(I)\neq0\}\setminus\{0\}$. The total graph of annihilating one-sided ideals of $R$, denoted by $\Omega(R)$, is a graph with the vertex set $A(R)^*$ and two distinct vertices $I$ and $J$ are adjacent if $\mathrm{l.ann}(I+J)\cup \mathrm{r.ann}(I+J)\neq0$. In this paper, we study the relations between the graph-theoretic properties of this graph and some algebraic properties of rings. We characterize all rings whose graphs are disconnected. Also, we study diameter, girth, independence number, domination number and planarity of this graph.

On Structure of Order Graphs Arising from Groups

Graph theory is a helpful tool in studying group properties. There has been a strong relationship between group (finite group) and graph theories for more than a century. The order graph of a finite group is the graph (undirected) whose vertices are the elements of the group and for any two distinct vertices H and K there is an edge from H to K , if and only if, order of H divides order of K (or vice-versa). In this paper, we focus on the interplay between the group-theoretic properties of G (finite group) and the graph-theoretic properties of Γ(G)(undirected graph).

Parikh word representability of bipartite permutation graphs

The class of Parikh word representable graphs were recently introduced. In this work, we further develop its general theory beyond the binary alphabet. Our main result shows that this class is equivalent to the class of bipartite permutation graphs. Furthermore, we study certain graph theoretic properties of these graphs in terms of the arity of the representing word.

Graphs Resemblance based Software Birthmarks through Data Mining for Piracy Control

The emergence of software artifacts greatly emphasizes the need for protecting intellectual property rights (IPR) hampered by software piracy requiring effective measures for software piracy control. Software birthmarking targets to counter ownership theft of software by identifying similarity of their origins. A novice birthmarking approach has been proposed in this paper that is based on hybrid of text-mining and graph-mining techniques. The code elements of a program and their relations with other elements have been identified through their properties (i.e., code constructs) and transformed into Graph Manipulation Language (GML). The software birthmarks generated by exploiting the graph theoretic properties (through clustering coefficient) are used for the classifications of similarity or dissimilarity of two programs. The proposed technique has been evaluated over metrics of credibility, resilience, method theft, modified code detection and self-copy detection for programs asserting the effectiveness of proposed approach against software ownership theft. The comparative analysis of proposed approach with contemporary ones shows better results for having properties and relations of program nodes and for employing dynamic techniques of graph mining without adding any overhead (such as increased program size and processing cost).

Centered Polygonal Lacunary Graphs: A Graph Theoretic Approach to p-Sequences of Centered Polygonal Lacunary Functions

This work is on the nature and properties of graphs which arise in the study of centered polygonal lacunary functions. Such graphs carry both graph-theoretic properties and properties related to the so-called p-sequences found in the study of centered polygonal lacunary functions. p-sequences are special bounded, cyclic sequences that occur at the natural boundary of centered polygonal lacunary functions at integer fractions of the primary symmetry angle. Here, these graphs are studied for their inherent properties. A ground-up set of planar graph construction schemes can be used to build the numerical values in p-sequences. Further, an associated three-dimensional graph is developed to provide a complementary viewpoint of the p-sequences. Polynomials can be assigned to these graphs, which characterize several important features. A natural reduction of the graphs original to the study of centered polygonal lacunary functions are called antipodal condensed graphs. This type of graph provides much additional insight into p-sequences, especially in regard to the special role of primes. The new concept of sprays is introduced, which enables a clear view of the scaling properties of the underling centered polygonal lacunary systems that the graphs represent. Two complementary scaling schemes are discussed.