Topological Generalizations of Graphs: Integrating Hypergraph and Superhypergraph Perspectives

Developments in graph colouring theory were motivated by the four-colour problem and Heawood's theorem. Both of these were originally formulated as map-colouring problems that can be expressed as colouring graphs embedded on surfaces. This chapter gives an overview of the abundance of results concerning the chromatic number of graphs that are embedded on surfaces . Introduction In 1852 Francis Guthrie asked whether the regions of every planar map can be coloured with four colours in such a way that no two regions of the map with common boundary receive the same colour. In effect, by duality Guthrie was asking whether every planar graph is 4-colourable. This easily stated problem became known as the famous four-colour problem. Attempts to solve it led to many important results in graph theory, but the problem itself remained unsolved for more than a century. It was finally answered in the positive by Appel and Haken [9], [10], [12]. More about its proof comes later in this chapter. For more than a century, the four-colour problem was one of the driving forces that led to developments in graph theory, and specifically graph colouring theory. Its generalization – the Heawood problem – was the main motivation for developments in topological graph theory (see [51] or [49]). Both of these were originally formulated as map-colouring problems, asking about colouring the faces of an embedded graph so that adjacent faces receive different colours. Every map-colouring problem can be expressed as a graph-colouring problem by considering the dual graph of the map. Its vertices correspond to the faces of the map, and two vertices are adjacent if the corresponding faces are adjacent on the surface. In this chapter we discuss the colouring of graphs embedded on surfaces. We start by describing some of the ideas in the proof of the four-colour theorem and give a proof of the list-colouring version of the five-colour theorem that is due to Thomassen.

Graph theoretical studies have been designed to investigate network topologies during life. Network science and graph theory methods may contribute to a better understanding of brain function, both normal and abnormal, throughout developmental stages. The degree to which childhood epilepsies exert a significant effect on brain network organisation and cognition remains unclear. The hypothesis suggests that the formation of abnormal networks associated with epileptogenesis early in life causes a disruption in normal brain network development and cognition, reflecting abnormalities in later life. Neurological diseases with onset during critical stages of brain maturation, including childhood epilepsy, may threaten this orderly neurodevelopmental process. According to the hypothesis that the formation of abnormal networks associated with epileptogenesis in early life causes a disruption in normal brain network development, it is then mandatory to perform a proper examination of children with new-onset epilepsy early in the disease course and a deep study of their brain network organisation over time. In regards, graph theoretical analysis could add more information. In order to facilitate further development of graph theory in childhood, we performed a systematic review to describe its application in functional dynamic connectivity using electroencephalographic (EEG) analysis, focussing on paediatric epilepsy.

The Laplacian is another important matrix associated with a graph, and the Laplacian spectrum is the spectrum of this matrix. We will consider the relationship between structural properties of a graph and the Laplacian spectrum, in a similar fashion to the spectral graph theory of previous chapters. We will meet Kirchhoff’s expression for the number of spanning trees of a graph as the determinant of the matrix we get by deleting a row and column from the Laplacian. This is one of the oldest results in algebraic graph theory. We will also see how the Laplacian can be used in a number of ways to provide interesting geometric representations of a graph. This is related to work on the Colin de Verdiere number of a graph, which is one of the most important recent developments in graph theory.

The workshop provided a venue for discussing several recent major developments in graph theory, with a primary focus on results concerning the notion of twin-width and hereditary properties of graphs. Both areas have seen a very rapid development recently, as reflected in the many results presented during the workshop. In addition to many interesting talks spanning the whole breadth of graph theory, the workshop also offered valuable collaboration opportunities, which also engaged early career researchers.

On edges crossing few other edges in simple topological complete graphs

This paper describes a programming system used in the teaching of two Computer Science courses at The Pennsylvania State University (7). The courses are; an upper level undergraduate-beginning graduate level course in graph theory, and a graduate level course in the theory of graphs and networks. As can be ascertained from the description of these courses they are not primarily what would normally be called programming courses. However, since they are computer science courses they do stress the computational aspects of graph and network theory and analysis. Past experiences have indicated that if a computational problem of any reasonable complexity is assigned in class the majority of the students focus an inordinate amount of their energies to the programming problems, and consequently fail to grasp the significant features of the theoretical problems involved. In addition, since graph and network theory encompasses not only computer science, but engineering and physical, and to a lesser extent, the social sciences, many of these students would be unduly burdened by writing many rather complex programs. On the other hand, teaching the computational aspects of graph theory without doing some programming is equally ludicrous. Thus, a default type solution of assigning two or three problems from a narrow set of problems was rapidly becoming an undesirable solution to this dilema. It was at this point that the concept of a dynamically growing programming system, as described in this paper, began to formulate itself.

We study the existence of edges having few crossings with the other edges in drawings of the complete graph (more precisely, in simple topological complete graphs). A topological graphT=(V,E) is a graph drawn in the plane with vertices represented by distinct points and edges represented by Jordan curves connecting the corresponding pairs of points (vertices), passing through no other vertices, and having the property that any intersection point of two edges is either a common end-point or a point where the two edges properly cross. A topological graph is simple, if any two edges meet in at most one common point.Let h=h(n) be the smallest integer such that every simple topological complete graph on n vertices contains an edge crossing at most h other edges. We show that Ω(n 3/2)≤ h(n) ≤ O(n 2/log1/4 n). We also show that the analogous function on other surfaces (torus, Klein bottle) grows as cn 2.

Transformer models have great potential in Graph Representation Learning (GRL) for efficiently scaling the learning process on large datasets and solving many challenges presented in Graph Neural Networks, e.g., oversmoothing and suspended animation. To represent each node of a graph, Transformer models as input usually take a node together with the node context, i.e., a set of other nodes that serve as learning context for the target node. However, current GRL Transformer models mainly consider the graph topology when selecting the node context for each target node. In this work, we demonstrate the important role of node features in selecting the node context. Specifically, we propose a hybrid approach for selecting node context that considers both the graph topology and the semantic similarities between node features. Through the empirical evaluations, we show the advantages of our hybrid node context selection method for a downstream classification task on various datasets compared to selection methods that only consider graph topology or semantic similarities. The best classification accuracy improvements of our proposed hybrid methods over the baseline methods on each dataset range from 0.77% to 6.05%.

Following this advice, we should look over the big questions, then doodle and sketch out some approaches. If you are an expert, then this is easy to do, but most people do not want to wait to become an expert before looking at interesting problems. Graph theory, our area of expertise, has many hard-to-solve questions. Some hark back to the recreational roots of the area yet still keep their mystery. These ‘‘acorns’’ can be planted on the backs of envelopes, on a blackboard, and over a coffee. Our goal is to collect some of these conjectures—arguably some of the most intriguing—in one place. We present ten conjectures in graph theory, and you can read about each one in at most ten minutes. As we live in the era of Twitter, all the conjectures we state are 140 characters or fewer (so ‘‘minute’’ here has a double meaning). We might even call these sketchy tweets, as we present examples for each conjecture that you can doodle on as you read. Hamming also references ambiguity: good researchers can work both on proving and disproving the same statement, so we approach the conjectures with an open mind. He also mentions that a good approach is to reframe the problem, and change the point of view. One example from Vizing’s Conjecture (which is discussed as our second-to-last conjecture in the following text), is the three-page paper [2] which, with a new way of thinking, reduced most of the published work of twenty years to a corollary of its main result! Given the size of modern graph theory, with its many smaller subfields (such as structural graph theory, random graphs, topological graph theory, graph algorithms, spectral graph theory, graph minors, and graph homomorphisms, to name a few), it would be impossible to list all, or even the bulk of the conjectures in the field. We are content instead to focus on a few family jewels, which have an intrinsic beauty and have provided some challenges for graph-theorists for at least two decades. There is something for everyone here, from undergraduate students taking their first course in graph theory, to seasoned researchers in the field. For additional reading on problems and conjectures in graph theory and other fields, see the Open Problem Garden maintained by IRMACS at Simon Fraser University [24]. We consider only finite and undirected graphs, with no multiple edges or loops (unless otherwise stated). We assume the reader has some basic familiarity with graphs and their terminology, including notions such as cycles, paths, complete graphs, complete bipartite graphs, vertex degrees, and connected graphs. We use the notation Cn for the cycle with n vertices, Pn for the path with n vertices, and Kn for the complete graph with n vertices. The complete bipartite graph with m and n vertices of the respective colors is denoted by Km,n. All the background we need can be found in any text in graph theory, such as those of Diestel [9] and West [41], or online (see for example [42]).

In this paper, we give sufficient conditions to have a complete synchronization of oscillators in connected undirected networks. The oscillators we are considering are not necessarily identical and the synchronization terms can be nonlinear. Many results in the literature deal with sufficient conditions insuring complete synchronization. This is a difficult problem since such conditions require to take into account the individual dynamics of the oscillators and also the graph topology. In this paper, we extend one of these results, the connection graph stability method, to nonlinear coupling functions and we also give an existence condition of trajectories of the oscillators. The sufficient conditions for synchronization presented in this paper are based on the study of a Lyapunov function and on the use of pseudometrics which enable us to link network dynamics and graph theory. These results are applied to a network of Chua’s oscillators and lead to an explicit condition insuring the complete synchronization of the oscillators.

The historical background of how graph theory emerged into world and gradually gained importance in different fields of study is very well stated in many books and articles. Some of the most important applications of graph theory can be seen in the field network theory. Its significance can be seen in some of the complex network systems in the field of biological system, ecological system, social systems as well as technological systems. In this paper, the basic concepts of graph theory in terms of network theory have been provided. The various network models like star network model, ring network model, and mesh network model have been presented along with their graphical representation. We have tried to establish the link between the models with the existing concepts in graph theory. Also, many application-based examples that links graph theory with network theory have been looked upon.KeywordsDegreeMatching indexOperatorsClusterStar network modelMesh network modelRing network model

Underwater wireless sensor networks (UWSNs) have become crucial for many different applications, such as marine pastures, which needs stratified aquaculture according to the habitat and range of activities of marine organisms. This type of application poses a significant challenge to network topology because strengthening the of the living conditions and collecting underwater data on organisms in different regions is necessary. In this paper, a new concept of subregional monitoring is proposed, and subregions with frequent activities of marine organisms are taken as the key areas, i.e., the areas of interest (AOIs). Then, SFRG, an energy-balanced and robust topology based on scale-free network and rigid graph theory, is proposed to strengthen the of the AOIs. The topology of a rigid graph is constructed in AOI, and the entire network is scale-free network with power-law degree distribution, when the rigid graph is regarded as a node. Furthermore, a transmission algorithm based on the SFRG is suggested to prolong the network lifetime. The algorithm guarantees that 90% of the received packets are from the AOIs. The simulation results demonstrate that the SFRG effectively prolongs the network lifetime and improves the network robustness, which provides a strong support in strengthening the of AOIs and balancing the energy consumption.

The field of mathematics that studies the relationship between algebraic structures and graphs is known as algebraic graph theory. It incorporates concepts from graph theory, which examines the characteristics and topology of graphs, with those from abstract algebra, which deals with algebraic structures such as groups, rings, and fields. If the vertex set of a graph G^ is fully made up of the zero divisors of the modular ring Zn, the graph is said to be a zero-divisor graph. If the products of two vertices are equal to zero under (modn), they are regarded as neighbors. Entropy, a notion taken from information theory and used in graph theory, measures the degree of uncertainty or unpredictability associated with a graph or its constituent elements. Entropy measurements may be used to calculate the structural complexity and information complexity of graphs. The first, second and second modified Zagrebs, general and inverse general Randics, third and fifth symmetric divisions, harmonic and inverse sum indices, and forgotten topological indices are a few topological indices that are examined in this article for particular families of zero-divisor graphs. A numerical and graphical comparison of computed topological indices over a proposed structure has been studied. Furthermore, different kinds of entropies, such as the first, second, and third redefined Zagreb, are also investigated for a number of families of zero-divisor graphs.

This paper presents a development of a novel, efficient, and robust tool to conduct evacuation from maritime ships using multi-commodity network theory and graph search techniques. MSC/1533 (IMO, 2016) presents guidelines to conduct simplified/quasi-advanced analysis methods for the evacuation analysis of ships. The IMO techniques describe a prescriptive approach and don’t consider path variabilities or uncertainties associated with the dynamic nature of the evacuation environment. To enhance the work presented by the IMO, a linear programming multi-commodity graphing solution has been developed and implemented in Excel. This model is based on network and graph theory and specifies basic passenger characteristics, simulates ships geometry and specifies path constraints. The simulation determines the shortest route to the nearest safe point/muster station. Monte Carlo is used to characterise model uncertainties in the simulation. Initial experiments indicate this technique is feasible. As a Systems Engineering tool, it can be used in trade-off studies to finalise ship layout design and identify path delay/choke points and can be used to rank evacuation performance of existing ship configurations.

Machine efficiency determined using graph and screw theories with application in robotics