Kinds Of Graphs Research Articles

Improved time complexity for spintronic oscillator ising machines compared to a popular classical optimization algorithm for the Max-Cut problem.

Solving certain combinatorial optimization problems like Max-Cut becomes challenging once the graph size and edge connectivity increase beyond a threshold, with brute-force algorithms which solve such problems exactly on conventional digital computers having the bottleneck of exponential time complexity. Hence currently, such problems are instead solved approximately using algorithms like Goemans-Williamson (GW) algorithm, run on conventional computers with polynomial time complexity. Phase binarized oscillators (PBOs), also often known as oscillator Ising machines, have been proposed as an alternative to solve such problems. In this paper, restricting ourselves to the combinatorial optimization problem Max-Cut solved on three kinds of graphs (Mobius Ladder, random cubic, Erdös Rényi) up to 100 nodes, we empirically show that computation time/time to solution (TTS) for PBOs (captured through Kuramoto model) grows at a much lower rate (logarithmically:O(log(N)), with respect to graph sizeN) compared to GW algorithm, for which TTS increases as square of graph size (O(N2)). However, Kuramoto model being a physics-agnostic mathematical model, this time complexity/ TTS trend for PBOs is a general trend and is device-physics agnostic. So for more specific results, we choose spintronic oscillators, known for their high operating frequency (in GHz), and model them through Slavin's model which captures the physics of their coupled magnetization oscillation dynamics. We thereby empirically show that TTS of spintronic oscillators also grows logarithmically with graph size (O(log(N)), while their accuracy is comparable to that of GW. So spintronic oscillators have improved time complexity over GW algorithm. For large graphs, they are expected to compute Max-Cut values much faster than GW algorithm, as well as other oscillators operating at lower frequencies, while maintaining the same level of accuracy.

Efficient Traffic Control Using Graph Theory: A Comprehensive Overview and Application

The use of graph theory in the area of traffic control is presented in-depth in this article. Urban transport networks have recurring difficulties with congestion, safety, and traffic efficiency. Graph theory offers a methodical and practical solution to these issues. We examine the fundamental ideas of graph theory in further detail, including its applicability to transportation networks and the many kinds of graphs that are employed in traffic control, including their static and dynamic performance. The article shows how graphical algorithms may be used to improve traffic management, including the shortest path method, flow optimisation, and traffic light synchronisation. Furthermore, we show the usefulness of graph theory in smart city traffic light synchronisation, public transit routing, incident management, and ITS integration through case studies and real-world applications. Data collecting, scalability, and the influence of autonomous cars are just a few of the difficulties and future trends in this sector that are highlighted. In conclusion, this study emphasises the crucial part that graph theory plays in raising transportation efficiency and safety. Its incorporation into traffic control systems holds the possibility of creating more flexible and responsive transport networks, enhancing urban sustainability.

On domination and 2-degree-packing numbers in graphs

A dominating set of a graph G is a set D ⊆ V (G) such that every vertex of G is either in D or is adjacent to a vertex in D. The domination number of G, γ(G), is the minimum order of a dominating set. The domination number has become an interesting research study on several kinds of graphs, and there are many papers related to this parameter and several variants of it. In this paper, we prove that any simple graph G with no isolated vertices satisfies γ(G) ≤ ν2(G) − 1, where ν2(G) is the maximum order of a subset R of edges of G such that any three edges from R do not have the same incident vertex. This new parameter is called the 2-degree-packing of G and it is studied in a more general context but with a different name as 2-packing number, see Araujo-Pardo et al. [Util. Math. 105 (2017) 317–336]. Also, in this paper, we give a characterization of simple connected graphs G satisfying γ(G) = ν2(G) − 1.

Threshold graphs maximise homomorphism densities

AbstractGiven a fixed graph $H$ and a constant $c \in [0,1]$ , we can ask what graphs $G$ with edge density $c$ asymptotically maximise the homomorphism density of $H$ in $G$ . For all $H$ for which this problem has been solved, the maximum is always asymptotically attained on one of two kinds of graphs: the quasi-star or the quasi-clique. We show that for any $H$ the maximising $G$ is asymptotically a threshold graph, while the quasi-clique and the quasi-star are the simplest threshold graphs, having only two parts. This result gives us a unified framework to derive a number of results on graph homomorphism maximisation, some of which were also found quite recently and independently using several different approaches. We show that there exist graphs $H$ and densities $c$ such that the optimising graph $G$ is neither the quasi-star nor the quasi-clique (Day and Sarkar, SIAM J. Discrete Math. 35(1), 294–306, 2021). We also show that for $c$ large enough all graphs $H$ maximise on the quasi-clique (Gerbner et al., J. Graph Theory 96(1), 34–43, 2021), and for any $c \in [0,1]$ the density of $K_{1,2}$ is always maximised on either the quasi-star or the quasi-clique (Ahlswede and Katona, Acta Math. Hung. 32(1–2), 97–120, 1978). Finally, we extend our results to uniform hypergraphs.

Binary programming formulations for the upper domination problem

We consider Upper Domination, the problem of finding the minimal dominating set of maximum cardinality. Very few exact algorithms have been described for solving Upper Domination. In particular, no binary programming formulations for Upper Domination have been described in literature, although such formulations have proved quite successful for other kinds of domination problems. We introduce two such binary programming formulations, and show that both can be improved with the addition of extra constraints which reduce the number of feasible solutions. We compare the performance of the formulations on various kinds of graphs, and demonstrate that (a) the additional constraints improve the performance of both formulations, and (b) the first formulation outperforms the second in most cases, although the second performs better for very sparse graphs. Also included is a short proof that the upper domination number of any generalized Petersen graph P(n, k) is equal to n.

Analysis of Metal Clusters Based on Graph-Theoretic Interpretation of the Lowest Occupied Molecular Orbital

Molecules can be regarded as a kind of graph with atoms as vertices and bonds as edges. Therefore, it is possible to discuss the physical properties and reactivity of various molecules by applying the ideas of graph theory and network theory. Such research has flourished in the field of chemical graph theory. In this manuscript, it is shown that the lowest occupied molecular orbital (LOMO) can be interpreted graph theoretically as eigenvector centrality. This is one measure of centrality that characterizes the importance or influence of a node on a network. As such, LOMO coefficient can be regarded as a manifestation of centrality in an aggregate of atoms, indicating which atom plays the most important role in that aggregate or has the greatest influence on the atom network. Using such properties of LOMO, an analysis of the network of metal atoms in metal clusters is performed. The predictability of the binding energies of the constituent atoms of the metal clusters using LOMO coefficients is discussed.

Weighted graphs with an application to returned sequence

Its well known that there are a limited number of method which associate group theory to graph theory. In this paper, we propose a new method and obtain a novel kind of graphs. Some important application like graph energy was established to the proposed graph.

Reliabilities for two kinds of graphs with smaller diameters

Two crucial parameters to be considered when evaluating a network’s reliability are its connectivity and diagnosability. The connectivity of a graph is defined as the smallest number of vertices that, if removed, would leave a disconnected or single-vertex graph. The diagnosability is the maximum number of faulty vertices that the system can correctly identify. In this paper, aim to obtain the reliability of various graphs (e.g. interconnection networks and data center networks), we propose two kinds of graphs (resp. networks)—matching recursive graphs (MRGs) and recursive networks based on connected graphs (RNCGs). We prove that these two kinds of graphs have smaller diameters. Furthermore, we study their reliabilities, including connectivities and diagnosabilities. Moreover, we also extend our results to many graphs.

ФУНКЦІОНУВАННЯ ІНФОГРАФІЧНИХ РЕСУРСІВ У ТЕКСТАХ НОВИН (НА МАТЕРІАЛІ ОНЛАЙН-НОВИН СЛУЖБ BBC І CNN)

The article highlights the use of infographics to effectively present the news material at BBC and CNN web pages. Infographic is a visual representation of information, data and knowledge. The infographic types that are widely used in news texts include maps, different kinds of graphs and charts such as bar/line/pie/column/bubble/gauge/stock/waterfall/flow charts; area/ hierarchy/circuit/tree diagrams; pictographs, timelines. Also, the symbiosis of infographic resources and additional graphic components (images, icons, geometric shapes, colour, numbers, and headers) was taken into account within the scopes of the current research. All of these allow controlling the infographics flow. Both verbal and nonverbal elements are considered to be crucially significant in creating the eye-catching and attention-grabbing infographic picture. Besides, the digital mode and colour play a significant role in presenting the information infographically. Numbers convey large amounts of information briefly and accurately and colour denotes emotionally valuable metaphorical meaning, stylistic and associative properties. The visual and the verbal can be integrated into a single syntagmatic unit. Verbal elements elucidate and decipher the nonverbal symbols, restate the message of the infographic image in a more precise way, distilling just one from the many possible meanings the image might have. Verbal items are represented by means of a single lexeme, a word combination or a simple sentence. Nonverbal components provide the verbal ones with the essential and highly informative visuals by showing the data, presenting many numbers in a small space, making large data sets coherent. They serve a reasonably clear purpose of description, exploration, tabulation and decoration. The visual elements serve as syntax identifying the participants by nonverbal means. Infographic is an effective and powerful tool to explain complex issues in a way that can quickly lead to insight and better understanding by making information easy to digest, educational, and engaging.

Parallel Learning of Dynamics in Complex Systems

Dynamics always exist in complex systems. Graphs (complex networks) are a mathematical form for describing a complex system abstractly. Dynamics can be learned efficiently from the structure and dynamics state of a graph. Learning the dynamics in graphs plays an important role in predicting and controlling complex systems. Most of the methods for learning dynamics in graphs run slowly in large graphs. The complexity of the large graph’s structure and its nonlinear dynamics aggravate this problem. To overcome these difficulties, we propose a general framework with two novel methods in this paper, the Dynamics-METIS (D-METIS) and the Partitioned Graph Neural Dynamics Learner (PGNDL). The general framework combines D-METIS and PGNDL to perform tasks for large graphs. D-METIS is a new algorithm that can partition a large graph into multiple subgraphs. D-METIS innovatively considers the dynamic changes in the graph. PGNDL is a new parallel model that consists of ordinary differential equation systems and graph neural networks (GNNs). It can quickly learn the dynamics of subgraphs in parallel. In this framework, D-METIS provides PGNDL with partitioned subgraphs, and PGNDL can solve the tasks of interpolation and extrapolation prediction. We exhibit the universality and superiority of our framework on four kinds of graphs with three kinds of dynamics through an experiment.

Scattering and Strebel graphs

We consider a special scattering experiment with n particles in \mathbb{R}^{1,n-3}ℝ1,n−3. The scattering equations in this set-up become the saddle-point equations of a Penner-like matrix model, where in the large nn limit, the spectral curve is directly related to the unique Strebel differential on a Riemann sphere with three punctures. The solutions to the scattering equations localize along different kinds of graphs, tuned by a kinematic variable. We conclude with a few comments on a connection between these graphs and scattering in the Gross-Mende limit.

Non-Heading Chinese Cabbage Database: An Open-Access Platform for the Genomics of Brassica campestris (syn. Brassica rapa) ssp. chinensis.

The availability of a high-quality genome sequence of Brassica campestris ssp. chinensis NHCC001 has paved the way for deep mining of genome data. We used the B. campestris NHCC001 draft genome to develop a comprehensive database, known as the non-heading Chinese cabbage database, which provides access to the B. campestris NHCC001 genome data. The database provides 127,347 SSR, from which 382,041 pairs of primers were designed. NHCCDB contains information on 105,360 genes, which were further classified into 63 transcription factor families. Furthermore, NHCCDB provides eight kinds of tools for biological or sequencing data analyses, including sequence alignment tools, functional genomics tools, comparative genomics tools, motif analysis tools, genome browser, primer design, and SSR analysis tools. In addition, eight kinds of graphs, including a box plot, Venn diagram, corrplot, Q-Q plot, Manhattan plot, seqLogo, volcano plot, and a heatmap, can be generated rapidly using NHCCDB. We have incorporated a search system for efficient mining of transcription factors and genes, along with an embedded data submit function in NHCCDB. We believe that the NHCCDB database will be a useful platform for non-heading Chinese cabbage research and breeding.

LWP-WL: Link weight prediction based on CNNs and the Weisfeiler–Lehman algorithm

We present a new technique for link weight prediction, the Link Weight Prediction Weisfeiler–Lehman (LWP-WL) method that learns from graph structure features and link relationship patterns. Inspired by the Weisfeiler–Lehman Neural Machine, LWP-WL extracts an enclosing subgraph for the target link and applies a graph labelling algorithm for weighted graphs to provide an ordered subgraph adjacency matrix into a neural network. The neural network contains a Convolutional Neural Network in the first layer that applies special filters adapted to the input graph representation. An extensive evaluation is provided that demonstrates an improvement over the state-of-the-art methods in several weighted graphs. Furthermore, we conduct an ablation study to show how adding different features to our approach improves our technique’s performance. Finally, we also perform a study on the complexity and scalability of our algorithm. Unlike other approaches, LWP-WL does not rely on a specific graph heuristic and can perform well in different kinds of graphs.

Exact Wiener Index of the Direct Product of a Path and a Wheel Graph

The direct product is one of the most important methods to construct large-scale graphs using existing small-scale graphs, and the topological structure parameters of the constructed large-scale graphs can be derived from small-scale graphs. For a simple undirected graph G , its Wiener index W G is defined as the sum of the distances between all different unordered pairs of vertices in the graph. Path is one of the most common and useful graphs, and it is found in almost all virtual and real networks; wheel graph is a kind of graph with good properties and convenient construction. In this paper, the exact value of the Wiener index of the direct product of a path and a wheel graph is given, and the obtained Wiener index is only derived from the orders of the two factor graphs.

A graph complexity measure based on the spectral analysis of the Laplace operator

In this work we introduce a concept of complexity for undirected graphs in terms of the spectral analysis of the Laplacian operator defined by the incidence matrix of the graph. Precisely, we compute the norm of the vector of eigenvalues of both the graph and its complement and take their product. Doing so, we obtain a quantity that satisfies two basic properties that are the expected for a measure of complexity. First, complexity of fully connected and fully disconnected graphs vanish. Second, complexity of complementary graphs coincide. This notion of complexity allows us to distinguish different kinds of graphs by placing them in a “croissant-shaped” region of the plane link density - complexity, highlighting some features like connectivity, concentration, uniformity or regularity and existence of clique-like clusters. Indeed, considering graphs with a fixed number of nodes, by plotting the link density versus the complexity we find that graphs generated by different methods take place at different regions of the plane. We consider some of the paradigmatic randomly generated graphs, in particular the Erdös-Rényi, the Watts-Strogatz and the Barabási-Albert models. Also, we place some particular, let us say deterministic, well known hand-crafted graphs, to wit, lattices, stars, hyper-concentrated and cliques-containing graphs. It is worthy noticing that these deterministic classical models of graphs depict the boundary of the croissant-shaped region. Finally, as an application to graphs generated by real measurements, we consider the brain connectivity graphs from two epileptic patients obtained from magnetoencephalography (MEG) recording, both in a baseline period and in ictal periods (epileptic seizures). In this case, our definition of complexity could be used as a tool for discerning between states, by the analysis of differences at distinct frequencies of the MEG recording.

A 2-connected graph in two dimensions containing 25 vertices with Gallai's Property

Graph theory recently considers as a modern field of mathematics and it was introduced by a great mathematician Leonhard Euler in 1735. Late then it is a flowered in the strong tool used in closely each area of science and nowadays it is most attractive and active area of mathematics research. Actually, graph theory is the study of graph, structure and molecules which have association in each other. It is very essential part of discrete mathematics which has been created with collaboration of nodes (vertices) and edges (lines or links), if a graph is connected with vertices directly by edges it is direct graph or a graph having symmetrically lines is also called direct graph or diagraph. In the studies of graph, we have investigated so many kinds of graph, Hypo-Hamiltonian is one of them. It is most important and it can be defined as the graph which hasn't Hamiltonian cycle but can be developed with the removing of single vertex form the developed graph. Lot of researchers have given the contribution in this research just like Naeem et al has worked on "A Two-Connected Graph with Gallai's Property" containing 12,18 and 25 vertices. In this research paper we have developed two different graphs consisting 25 Eulerian graph which has highest cycle and path order C( G) — 24 and P( G) — 25. Other graph is in 3D under cycle which contains 23 nodes.

Structure of intersection graphs

<p style="text-align: left;" dir="ltr"> Let <em>G</em> be a finite group and let <em>N</em> be a fixed normal subgroup of <em>G</em>. In this paper, a new kind of graph on <em>G</em>, namely the intersection graph is defined and studied. We use <img src="/public/site/images/ikhsan/equation.png" alt="" width="6" height="4" /> to denote this graph, with its vertices are all normal subgroups of <em>G</em> and two distinct vertices are adjacent if their intersection in <em>N</em>. We show some properties of this graph. For instance, the intersection graph is a simple connected with diameter at most two. Furthermore we give the graph structure of <img src="/public/site/images/ikhsan/equation_(1).png" alt="" width="6" height="4" /> for some finite groups such as the symmetric, dihedral, special linear group, quaternion and cyclic groups. </p>

Exploring actionable visualizations for environmental data: Air quality assessment of two Belgian locations

Organizations collect an ever-increasing amount of data concerning environmental parameters. Non-experts may be confronted with an information overload, making the data meaningless. A way to circumvent this problem is visualizing the trends in the pollution concentrations measured over time. However, non-experts do not have a mental model to derive air quality information from displayed concentration profiles. Therefore, a large fraction of the stakeholders remains unable to read/interpret such data effectively. To improve communication with stakeholders, we superposed health risk information from 9 different Air Quality Indices (AQIs) on different kinds of graphs. The visualization methods are applied on data collected by the Belgian Environment Agency from two monitoring stations located in contrasting regions, Ghent and Vielsalm. Supplementary, spatially distributed pollution is shown using data collected from Sentinel-5p satellite. Despite some limitations of the AQIs, the applied visualizations methods successfully translate the data obtained into actionable information.

Harmonious graphs from α-trees

Two of the most studied graph labelings are the types of harmonious and graceful. A harmonious labeling of a graph of size m and order n , is an injective assignment of nonnegative integers smaller than m , such that the weights of the edges, which are defined as the sum of the labels of the end-vertices, are distinct consecutive integers after reducing modulo m . When n = m + 1 , exactly two vertices of the graph have the same label. An α -labeling of a tree of size m is a bijective assignment of nonnegative integers, not larger than m, such that the labels on one stable set are smaller than the labels on the other stable set, and the weights of the edges, which are defined as the absolute difference of the labels of the end-vertices, are all distinct; this is the most restrictive type of graceful labeling. Even when these labelings are significantly different in their definitions of the weight, for certain kinds of graphs, there is a deep connection between harmonious and α -labelings. We present new families of harmoniously labeled graphs built on α -labeled trees. Among these new results there are three families of trees, the k th power of the path P n , the join of a graph G and t K 1 where G is a graph that admits a more restrictive type of harmonious labeling and its order is different of its size by at most one unit. We also prove the existence of two families of disconnected harmonius graphs: K n , m ∪ K 1, m − 1 and G ∪ T , where G is a unicyclic graph and T is a tree built with α -trees. In addition, we show that almost all trees admit a harmonious labeling.

Fast Near-Optimal Path Planning on State Lattices with Subgoal Graphs

Subgoal graphs can be constructed on top of graphs during a preprocessing phase to speed-up shortest path queries. They have an undominated query-time/memory trade-off in the Grid-Based Path Planning Competitions. While grids are useful for path planning, other kinds of graphs, such as state lattices, have to be used for motion planning. While state lattices are regular graphs like grids, subgoal graphs improve query times by a much smaller factor on state lattices than on grids. In this paper, we present a new version of subgoal graphs that forfeits its optimality guarantee for smaller query times. It guarantees completeness, and our experimental results on state lattices suggest that it can find paths that are close to optimal.