Definition Of Graph Research Articles

Entropies Via Various Molecular Descriptors of Layer Structure of H3BO3

Entropy is essential. Entropy is a measure of a system’s molecular disorder or unpredictability, since work is produced by organized molecular motion. Entropy theory offers a profound understanding of the direction of spontaneous change for many commonplace events. A formal definition of a random graph exists. It deals with relational data’s probabilistic and structural properties. The lower-order distribution of an ensemble of attributed graphs may be used to describe the ensemble by considering it to be the results of a random graph. Shannon’s entropy metric is applied to represent a random graph’s variability. A structural or physicochemical characteristic of a molecule or component of a molecule is known as a molecular descriptor. A mathematical correlation between a chemical’s quantitative molecular descriptors and its toxicological endpoint is known as a QSAR model for predictive toxicology. Numerous physicochemical, toxicological, and pharmacological characteristics of chemical substances help to foretell their type and mode of action. Topological indices were developed some 150 years ago as an alternative to the Herculean, and arduous testing is needed to examine these features. This article uses various computational and mathematical techniques to calculate atom–bond connectivity entropy, atom–bond sum connectivity entropy, the newly defined Albertson entropy using the Albertson index, and the IRM entropy using the IRM index. We use the subdivision and line graph of the H3BO3 layer structure, which contains one boron atom and three oxygen atoms to form the chemical boric acid.

BILANGAN DOMINASI-LOKASI HASIL KALI SISIR GRAF LINTASAN DENGAN BEBERAPA GRAF REGULER

To all the paper, all graph is connected, simple, undirected, and finite. A set vertex W of is called the domination set if every vertex for every is adjacent to some vertex The minimum cardinality of a location-dominated set is called the location-dominated number, which is denoted by λ(G). In this research, we observe to the comb product of a path graph to regular graphs. If noticed to the definition of a regular graph, then obtained a complete graph is graph with order . So in the results of this research we start by looking for the location-dominated number of the comb product of a path graph with a regular graph where the regular graph is a , end which each one has , , dan where order end .

상호연결망 NSEP Graph와 Circle Graph에 대한 Mathematica 코딩 기반의 시각화 분석

Research on interconnection networks is continuously needed to improve the performance of parallel processing computers. Using Mathematica, ambiguous parts of the graph can be specifically implemented and tested, and efficient and accurate analysis is possible by reducing meaningless guesses or time. It is also reusable, making it convenient to analyze the additional properties of the graph later. The interconnection network NSEP has a graph with one edge added to the SEP graph with a fixed degree 3, and the CG graph has a degree 3. NSEP graphs and CG graphs require further research in addition to the properties analyzed to date. To this end, three fixed degree graphs, including SEP, the leading graph of NSEP, were designed and new properties were analyzed in Mathematica. NSEP, an interconnected network with a fixed degree, has a graph with one edge added to the SEP graph with a fixed degree 3, and the CG graph has a degree 3. A graph with a fixed degree has an advantage in that the degree does not increase as the dimension increases, but has a disadvantage in that the number of nodes increases rapidly.
 Compared to the existing graph definition, the diameter of SEP and NSEP analyzed in Mathematica was smaller. The diameter of the SEP is 1/8(9n^2-22n+24), and the diameter of the NSEP is 2/3n^2-3/2n+1. Since CG has the same result as the existing diameter A, it can be seen that the optimal routing algorithm is proven.
 This study showed how to generate interconnection networks using Mathematica coding and presented the results of efficiently analyzing the discrete structure by exploring the graph properties as well as providing visualization methods that can be used as field-based teaching contents for discrete mathematics.

Partitioning of urban networks with polycentric congestion pattern for traffic management policies: Identifying protected networks

AbstractLarge‐scale urban networks are usually loaded heterogeneously with a polycentric congestion pattern, resulting in a highly scattered network macroscopic fundamental diagram (NMFD or MFD). Thus, researchers have tried to partition city networks into homogeneous subzones. In this study, a six‐step partitioning algorithm is proposed. The framework allows the NMFD information to be used. It combines traffic variables and geographic connectivity to obtain partitions. The framework includes graph definition, data preprocessing, feature handling, clusters and partitions identification, and boundaries reshaping. Tests on a simplified grid network and the city of the Melbourne road network demonstrate the suitability of the framework for characterizing the traffic states by the partitions. The framework on the missing data scenarios and monocentric and polycentric traffic concentrations scenarios, as well as applying multiple data types, has been challenged. Thereafter, parameter impact analysis demonstrates that manipulation of the parameters enables users to find the desired partitions. Last but not least, a comparison with an existing method also implies the uniqueness and efficiency of the developed framework.

Introduction to the Class of Prefractal Graphs

Fractals are already firmly rooted in modern science. Research continues on the fractal properties of objects in physics, chemistry, biology and many other scientific fields. Fractal graphs as a discrete representation are used to model and describe the structure of various objects and processes, both natural and artificial. The paper proposes an introduction to prefractal graphs. The main definitions and notation are proposed—the concept of a seed, the operations of processing a seed, the procedure for generating a prefractal graph. Canonical (typical) and non-canonical (special) types of prefractal graphs are considered separately. Important characteristics are proposed and described—the preservation of adjacency of edges for different ranks in the trajectory. The definition of subgraph-seeds of different ranks is given separately. Rules for weighting a prefractal graph by natural numbers and intervals are proposed. Separately, the definition of a fractal graph as infinite is given, and the differences between the concepts of fractal and prefractal graphs are described. At the end of the work, already published works of the authors are proposed, indicating the main backlogs, as well as a list of directions for new research. This work is the beginning of a cycle of works on the study of the properties and characteristics of fractal and prefractal graphs.

Review of Shortest Path Algorithm

Shortest path algorithm. Graphs are an example of non-linear data structure. A graph is a collection of nodes which are connected by edges. The definition of graph G = (V, E) is basically a collection of vertices and edges. Graphs can be classified on the basis of types of edges. Directed graphs have each of the edges directed which means the edges connecting the two nodes defines the way it is connected from and to. On the other side, undirected graphs have edges which have no direction. The edges of a graph have weights which are associated with it. The weight of an edge can be thought as the cost of the edge. Let’s assume there are two vertices representing two cities, then the weight of the edge between the vertices may represent the distance between the cities. Given a given graph and a particular node, we can find a path of least total weight from that node to other vertices of the graph. The total weight of the path will be the sum of the weights of the edges. Graphs can be used in real life to find the shortest path between two destinations, used in social networking sites like facebook and the world wide web where the web pages are represented by the nodes. Dijkstra’s Algorithm, Floyd – Warshall, Bellman Ford Algorithm, Johnson’s algorithm, A* search algorithm are some of the shortest path algorithms.

L-graph Automata And Some Applications

Abstract This paper defines a novel graph constructed on a residuated lattice using a specific definition of the fuzzy graph (called $L$-graph). These types of graphs also have applications in libraries, pharmacies and machine facilities. Among these applications, this research presents the $L$-graph automaton related to the $L$-graph constructed by a minimum zero forcing set. The $L$-graph automaton is used to find efficient searching for diseases that have the most similar symptoms. It has been demonstrated that if two $L$-graphs are isomorphic, then the related $L$-graph automata are isomorphic but not necessarily vice versa. New notions, $L^{\prime\prime}$-graph automata, pseudo-isomorphism and self-sufficiency are proved. Besides, by taking advantage of the properties of the residuated lattice, it has been proven that the two $L$-graph automata are equivalent if and only if the $L^{\prime}$-graph automata contracted of $L$-graph automata are equivalent. Ultimately, in light of the above, some related theorems are proved and several examples are provided to illustrate these new notions.

ODD HARMONIOUS LABELING ON SOME STRING GRAPH CLASSES

A graph with the labeling properties of odd harmonic is called an odd harmonious graph. The purpose of this research was to get labeling properties of odd harmonic on the class of string graphs. The research used was a qualitative research method. The result of the research was that the definition and construction of a string graph, the union of a string graph, and the multiple string graph are obtained. Furthermore, it has been proved that a string graph, the union of a string graph, and the multiple string graph is an odd harmonious graph.

Some families of integral mixed graphs

A mixed graph Gˆ is a graph where two vertices can be connected by an edge or by an arc (directed edge). The adjacency matrix, Aˆ(Gˆ), of a mixed graph has rows and columns indexed by the set of vertices of Gˆ, being its {u,v}-entry equal to 1 (respectively, −1) if the vertex u is connected by an edge (respectively, an arc) to the vertex v, and 0 otherwise. These graphs are called integral mixed graphs if the eigenvalues of its adjacency matrix are integers. In this paper, symmetric block circulant matrices are characterized, and as a consequence, the definition of a mixed graph to be a block circulant graph is presented. Moreover, using this concept and the concept of a g-circulant matrix, the construction of a family of undirected graphs that are integral block circulant graphs is shown. These results are extended using the notion of H-join operation to characterize the spectrum of a family of integral mixed graphs. Furthermore, a new binary operation called mixed asymmetric product of mixed graphs is introduced, and the notions of joining by arcs and joining by edges are used, allowing us to obtain a new integral mixed graph from two original integral mixed graphs.

Non-Braid Graphs of Ring Zn

The research in graph theory has been widened by combining it with ring. In this paper, we introduce the definition of a non-braid graph of a ring. The non-braid graph of a ring R, denoted by YR, is a simple graph with a vertex set R\B(R), where B(R) is the set of x in R such that xyx=yxy for all y in R. Two distinct vertices x and y are adjacent if and only if xyx not equal to yxy. The method that we use to observe the non-braid graphs of Zn is by seeing the adjacency of the vertices and its braider. The main objective of this paper is to prove the completeness and connectedness of the non-braid graph of ring Zn. We prove that if n is a prime number, the non-braid graph of Zn is a complete graph. For all n greater than equal to 3, the non-braid graph of Zn is a connected graph.

SMGen: A Generator of Synthetic Models of Biochemical Reaction Networks

Several software tools for the simulation and analysis of biochemical reaction networks have been developed in the last decades; however, assessing and comparing their computational performance in executing the typical tasks of computational systems biology can be limited by the lack of a standardized benchmarking approach. To overcome these limitations, we propose here a novel tool, named SMGen, designed to automatically generate synthetic models of reaction networks that, by construction, are characterized by relevant features (e.g., system connectivity and reaction discreteness) and non-trivial emergent dynamics of real biochemical networks. The generation of synthetic models in SMGen is based on the definition of an undirected graph consisting of a single connected component that, generally, results in a computationally demanding task; to speed up the overall process, SMGen exploits a main–worker paradigm. SMGen is also provided with a user-friendly graphical user interface, which allows the user to easily set up all the parameters required to generate a set of synthetic models with any number of reactions and species. We analysed the computational performance of SMGen by generating batches of symmetric and asymmetric reaction-based models (RBMs) of increasing size, showing how a different number of reactions and/or species affects the generation time. Our results show that when the number of reactions is higher than the number of species, SMGen has to identify and correct a large number of errors during the creation process of the RBMs, a circumstance that increases the running time. Still, SMGen can generate synthetic models with hundreds of species and reactions in less than 7 s.

Lipschitz graphs and currents in Heisenberg groups

AbstractThe main result of the present article is a Rademacher-type theorem for intrinsic Lipschitz graphs of codimension$k\leq n$in sub-Riemannian Heisenberg groups${\mathbb H}^{n}$. For the purpose of proving such a result, we settle several related questions pertaining both to the theory of intrinsic Lipschitz graphs and to the one of currents. First, we prove an extension result for intrinsic Lipschitz graphs as well as a uniform approximation theorem by means of smooth graphs: both of these results stem from a new definition (equivalent to the one introduced by B. Franchi, R. Serapioni and F. Serra Cassano) of intrinsic Lipschitz graphs and are valid for a more general class of intrinsic Lipschitz graphs in Carnot groups. Second, our proof of Rademacher’s theorem heavily uses the language of currents in Heisenberg groups: one key result is, for us, a version of the celebrated constancy theorem. Inasmuch as Heisenberg currents are defined in terms of Rumin’s complex of differential forms, we also provide a convenient basis of Rumin’s spaces. Eventually, we provide some applications of Rademacher’s theorem including a Lusin-type result for intrinsic Lipschitz graphs, the equivalence between${\mathbb H}$-rectifiability and ‘Lipschitz’${\mathbb H}$-rectifiability and an area formula for intrinsic Lipschitz graphs in Heisenberg groups.

Dominating sets in random graphs

This work presents some random graphs with their dominating sets found by us. After giving an overview of the literature related to the topic and the route of scientific development leading to the definition of the random graphs, the logical process leading to the dominating sets of random graphs has been presented.

Characterizations of line graphs in signed and gain graphs

We generalize three classical characterizations of line graphs to line graphs of signed and gain graphs: the Krausz’s characterization, the van Rooij and Wilf’s characterization and the Beineke’s characterization. In particular, we present a list of forbidden gain subgraphs characterizing the class of gain-line graphs. In the case of a signed graph whose underlying graph is a line graph, this list consists of exactly four signed graphs. Under the same hypothesis, we prove that a signed graph is the line graph of a signed graph if and only if its eigenvalues are either greater than −2, or less than 2, depending on which particular definition of line graph is adopted.

Upper whole domination in a graph

The new parameters of domination in graphs are being initiated that are the upper whole domination and its inverse. Many properties and theorems about the upper whole dominations number have been discussed. Also, it is determined to certain graphs as a cycle, complete, path, wheel, and complete bipartite with some properties. Finally, most properties of inverse upper domination number have been studied. Furthermore, determined this definition of certain graphs are been mentioned above.

A Graph Theory of Rook Placements

Two boards are rook equivalent if they have the same number of non-attacking rook placements for any number of rooks. Define a rook equivalence graph on an equivalence class of Ferrers boards by specifying that two boards are connected by an edge if you can obtain one of the boards by moving squares in the other board out of one column and into a single other column. Given such a graph, we characterize which boards will yield connected graphs. We also provide some cases where common graphs will or will not be the graph for some set of rook equivalent Ferrers boards. Finally, we extend this graph definition to the m-level rook placement generalization developed by Briggs and Remmel. This yields a graph on the set of rook equivalent singleton boards, and we characterize which singleton boards give rise to a connected graph.

The power graph of a torsion-free group of nilpotency class 2

The directed power graph $$\vec {\mathcal {G}}( G)$$ of a group G is the simple digraph with vertex set G in which $$x\rightarrow y$$ if y is a power of x, and the power graph is the underlying simple graph $$\mathcal {G}( G)$$ . In this paper, three versions of the definition of the power graph are discussed, and it is proved that the power graph by any of the three versions of the definition determines the other two up to isomorphism. It is also proved that if G is a torsion-free group of nilpotency class 2 and if H is a group such that $$\mathcal {G}( H)\cong \mathcal {G}( G)$$ , then G and H have isomorphic directed power graphs, which was an open problem proposed by Cameron, Guerra and Jurina [9].

On generalized zero-divisor graphs of a non-commutative ring with respect to an ideal

Let R be a non-commutative ring, and I be an ideal of R. In this paper, we generalize the definition of the zero-divisor graph of R with respect to I, and define several generalized zero-divisor graphs of R with respect to I. In this paper, we investigate the ring-theoretic properties of R and the graph-theoretic properties of all the generalized zero-divisor graphs. We study some basic properties of these generalized zero-divisor graphs related to the connectedness, the diameter and the girth. We also investigate some properties of these generalized zero-divisor graphs with respect to primal ideals.

Hamming-shifting graph of genomic short reads: Efficient construction and its application for compression.

Graphs such as de Bruijn graphs and OLC (overlap-layout-consensus) graphs have been widely adopted for the de novo assembly of genomic short reads. This work studies another important problem in the field: how graphs can be used for high-performance compression of the large-scale sequencing data. We present a novel graph definition named Hamming-Shifting graph to address this problem. The definition originates from the technological characteristics of next-generation sequencing machines, aiming to link all pairs of distinct reads that have a small Hamming distance or a small shifting offset or both. We compute multiple lexicographically minimal k-mers to index the reads for an efficient search of the weight-lightest edges, and we prove a very high probability of successfully detecting these edges. The resulted graph creates a full mutual reference of the reads to cascade a code-minimized transfer of every child-read for an optimal compression. We conducted compression experiments on the minimum spanning forest of this extremely sparse graph, and achieved a 10 − 30% more file size reduction compared to the best compression results using existing algorithms. As future work, the separation and connectivity degrees of these giant graphs can be used as economical measurements or protocols for quick quality assessment of wet-lab machines, for sufficiency control of genomic library preparation, and for accurate de novo genome assembly.

Exploring the Blocks World state space

AbstractIn this article we study and evaluate combinatorial algorithms for exploring state spaces of prototypical artificial intelligence problems that are represented by very large state space graphs. We are interested in the modeling and analysis of such graphs, including their visualization and quantification. We consider Blocks World as a prototype artificial intelligence problem that is often employed to introduce problem solving strategies using searching, planning, and reasoning algorithms. Our results are mathematical models and combinatorial algorithms supporting the concise definition of the Blocks World state space graph, as well as the exact evaluation of several metrics defined on this graph, including its number of states, its average number of stacks per state, its number of transitions, and its average branching factor. We also present experimental results supporting the effectiveness and efficiency of our proposed algorithms.