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For the formation and obstacle avoidance challenges of UAVs (unmanned aerial vehicles) in complex scenarios, this paper proposes an improved collaborative strategy based on APF (artificial potential field). This strategy combines graph theory, the Leader–Follower method, and APF. Firstly, we used graph theory to design formation topology and dynamically adjust the distances between UAVs in real time. Secondly, we introduced APF to avoid obstacles in complicated environments. This algorithm innovatively integrates the Leader–Follower formation method. The design of this attractive field is replaced by the leader's attraction to the followers, overcoming the problem of unreachable targets in APF. Meanwhile, the introduced Leader–Follower mode reduces information exchange within the swarm, realizing a more efficient "few controlling many" paradigm. Afterwards, we incorporated rotational force to assist the swarm in breaking free from local minima. Ultimately, the stability of the integrated formation strategy was demonstrated using Lyapunov functions. The feasibility and effectiveness of the proposed strategy were validated across multiple platforms. [ABSTRACT FROM AUTHOR]

In 1975, Symons [[5]] introduced a subsemigroup of HT ht , defined as HT ht , for a nonempty subset I Y i of I X i , determining all its automorphisms. The first paper [[21]] introduces a construction of a new graph associated with a semihypergroup, using the fundamental relation HT ht . In particular, it is shown that the Cayley graphs generated by transposition trees on the set HT ht are HT ht -extendable and their extendability number is HT ht for any integer HT ht . Connections between algebraic structure theory and graph theory have been established in order to solve open problems in one theory with the help of the tools existing in the other, emphasizing the remarkable properties of one theory with techniques involving the second. [Extracted from the article]

Many properties of the structure and dynamics of complex networks derive from the characteristics of the spectrum of the associated Laplacian matrix, specifically from the set of its eigenvalues. In this paper, we show that there exist graphs for which the ratio between the length of the spectrum (that is, the difference between the largest and smallest eigenvalues of the Laplacian matrix) and its spread (the difference between the second smallest eigenvalue and the smallest one) is equal to the golden ratio. We call such graphs Golden Laplacian Graphs (GLG). In this paper, we first find all such graphs with a number of nodes n ≤ 10 . We then prove several graph-theoretic and algebraic properties that characterize these graphs. These graphs prove to be extremely robust, as they have large vertex and edge connectivity along with a large isoperimetric constant. Finally, we study the synchronization properties of GLGs, showing that they are among the top synchronizable graphs of the same size. Therefore, GLGs represent very good candidates for engineering and communication networks. [ABSTRACT FROM AUTHOR]

This is a noticeably short biography and introductory paper on multiplier Hopf algebras. It delves into questions regarding the significance of this abstract construction and the motivation behind its creation. It also concerns quantum linear groups, especially the coordinate ring of Mq(n) and the observation that K [Mq(n)] is a quadratic algebra, and can be equipped with a multiplier Hopf ∗-algebra structure in the sense of quantum permutation groups developed byWang and an observation by Rollier–Vaes. In our next paper, we will propose the study of multiplier Hopf graph algebras. The current paper can be viewed as a precursor to this upcoming work, serving as a crucial intermediary bridging the gap between the abstract concept of multiplier Hopf algebras and the well-developed field of graph theory, thereby establishing connections between them! This survey review paper is dedicated to the 78th birthday anniversary of Professor Alfons Van Daele. [ABSTRACT FROM AUTHOR]

Various methods are used when building bankruptcy prediction models. New sophisticated methods that are already used in other scientific fields can also be applied in this area. Graph theory provides a powerful framework for analyzing and visualizing complex systems, making it a valuable tool for assessing the sustainability and financial health of businesses. The motivation for the research was the interest in the application of this method rarely applied in predicting the bankruptcy of companies. The paper aims to propose an improved dynamic bankruptcy prediction model based on graph theoretical modelling. The dynamic model considering the causality relation between financial features was built for the period 2015–2021. Financial features entering the model were selected with the use of Domain knowledge approach. When building the model, the weights of partial permanents were proposed to determine their impact on the final permanent and the algorithm for the optimalisation of these weights was established to obtain the best performing model. The outcome of the paper is the improved dynamic graph theoretical model with a good classification accuracy. The developed model is applicable in the field of bankruptcy prediction and is an equivalent sophisticated alternative to already established models. [ABSTRACT FROM AUTHOR]

In this paper, three sorts of network indices for the weighted three-layered graph are studied through the methods of graph spectra theory combined with analysis methods. The concept of union of graphs are applied to design two sorts of weighted layered multi-fan composed graphs, and the accurate mathematical expressions of the network indices are obtained through the derivations of Laplacian spectra; furthermore, the asymptotic properties are also derived. We find that when the cardinalities of the vertices on a sector-edge-link tend to infinity, the indices of FONC and EMFPT are irrelevant with the number of copies of the fan-substructure based on the considered graph framework. [ABSTRACT FROM AUTHOR]

The problem of finding the maximum weighted cycle in a directed graph map to solve optimization problems is NP -hard, implying that approaches in classical computing are inefficient. Here, Quantum computing might be a promising alternative. Many current approaches to the quantum computer are based on a Quadratic Unconstrained Binary Optimization (QUBO) problem formulation. This paper develops four different QUBO approaches to this problem. The first two take the starting vertex and the number of vertices used in the cycle as given, while the latter two loosen the second assumption of knowing the size of the cycle. A QUBO formulation is derived for each approach. Further, the number of binary variables required to encode the maximum weighted cycle problem with one or both assumptions for the respective approach is made explicit. The problem is motivated by finding the maximum weighted debt cycle in a debt graph. This paper compares classical computing versus currently available (hybrid) quantum computing approaches for various debt graphs. For the classical part, it investigated the Depth-First-Search (DFS) method and Simulated Annealing. For the (hybrid) quantum approaches, a direct embedding on the quantum annealer and two types of quantum hybrid solvers were utilized. Simulated Annealing and the usage of the hybrid CQM (Constrained Quadratic Model) had promising functionality. The DFS method, direct QPU, and hybrid BQM (Binary Quadratic Model), on the other hand, performed less due to memory issues, surpassing the limit of decision variables and finding the right penalty values, respectively. [ABSTRACT FROM AUTHOR]

In our earlier work, we suggested an axiomatic framework for deducing the fundamental entities which constitute the building block of the elementary particles in physics. The basic concept of this theory, named apriorics, is the ontological structure (OS)—an undirected simple graph satisfying specified conditions. The vertices of this graph represent the fundamental entities (FEs), its edges are binary compounds of the FEs (which are the fundamental bosons and fermions), and the structures constituting more than two connected vertices are composite particles. The objective of this paper is to focus the attention on several mathematical theorems and ideas associated with such graphs of order n, including their enumeration, showing what is the information content of apriorics. [ABSTRACT FROM AUTHOR]

Moran's I (Moran's coefficient) is one of the most prominent measures of spatial autocorrelation. It is well known that Moran's I has a representation that is similar to a Fourier series and is therefore useful for characterizing spatial data. However, the representation needs to be modified. This paper contributes to the literature by showing the necessary modification and presenting some further results. In addition, we provide the required MATLAB/GNU Octave and R user-defined functions. [ABSTRACT FROM AUTHOR]

Graph convolutional network (GCN) architecture is the basis of many neural networks and has been widely used in processing graph-structured data. When dealing with large and sparse data, deeper GCN models are often required. However, the models suffer from performance degradation as the number of layers increases. The mainstream attribution of the current research is over-smoothing, and there are also gradient vanishing, training difficulties, etc., so a consensus cannot be reached. In this paper, we theoretically analyze the degradation problem by adopting spectral graph theory to globally consider the propagation and transformation components of the GCN architecture, and conclude that the over-smoothing problem caused by the propagation matrices is not the key factor for performance degradation. Afterwards, in addition to using conventional experimental methods, we proposed an experimental analysis strategy under the guidance of random matrix theory to analyze the singular value distribution of the model weight matrix. We concluded that the key factor leading to the degradation of model performance is the transformation component. In the context of a lack of consensus on the problem of model performance degradation, the paper proposes a systematic analysis strategy, as well as theoretical and empirical evidence. [ABSTRACT FROM AUTHOR]

In this paper, we consider the adjacent vertex distinguishing proper edge coloring (for short, AVDPEC) and the adjacent vertex distinguishing total coloring (for short, AVDTC) of a fuzzy graph. Firstly, this paper describes the development process, the application areas, and the existing review research of fuzzy graphs and adjacent vertex distinguishing coloring of crisp graphs. Secondly, we briefly introduce the coloring theory of crisp graphs and the related theoretical basis of fuzzy graphs, and add some new classes of fuzzy graphs. Then, based on the α -cuts of fuzzy graphs and distance functions, we give two definitions of the AVDPEC of fuzzy graphs, respectively. A lower bound on the chromatic number of the AVDPEC of a fuzzy graph is obtained. With examples, we show that some results of the AVDPEC of a crisp graph do not carry over to our set up; the adjacent vertex distinguishing chromatic number of the fuzzy graph is different from the general chromatic number of a fuzzy graph. We also give a simple algorithm to construct a (d , f) -extended AVDPEC for fuzzy graphs. After that, in a similar way, two definitions of the AVDTC of fuzzy graphs are discussed. Finally, the future research directions of distinguishing coloring of fuzzy graphs are given. [ABSTRACT FROM AUTHOR]

This paper initiates the study of the mathematical aspects of the ad-hoc Lanzhou index. If G is a graph with the vertex set { x 1 , ... , x n } , then the ad-hoc Lanzhou index of G is defined by L z ˜ (G) = ∑ i = 1 n d i (n − 1 − d i) 2 , where d i represents the degree of the vertex x i . Several identities for the ad-hoc Lanzhou index, involving some existing topological indices, are established. The problems of finding graphs with the extremum values of the ad-hoc Lanzhou index from the following sets of graphs are also attacked: (i) the set of all connected ξ -cyclic graphs of a fixed order, (ii) the set of all connected molecular ξ -cyclic graphs of a fixed order, (iii) the set of all graphs of a fixed order, and (iv) the set of all connected molecular graphs of a fixed order. [ABSTRACT FROM AUTHOR]

The frequency and severity of natural disasters is surging, posing an urgent need for robust communication network infrastructure that is capable of withstanding these events. In this paper, we present a groundbreaking graph-theoretic system designed to evaluate and enhance network resilience in the face of natural disasters. Our solution harnesses the power of topological robustness metrics, integrating real-time weather data, geographic information, detailed network topology data, advanced resilience algorithms, and continuous network monitoring. The proposed scheme considers four major real-world U.S.-based network providers and evaluates their physical topologies against two major hurricanes. Our novel framework quantifies the important characteristics of network infrastructure; for instance, AT&T is identified to have fared better against Hurricane Ivan (57.98 points) than Hurricane Katrina (39.17 points). We not only provide current insights into network infrastructure resilience, but also uncover valuable findings that shed light on the performance of backbone U.S. networks during hurricanes. Furthermore, our findings provide actionable insights to enrich the overall survivability and functionality of communication networks, mitigating the adverse impacts of natural disasters on communication systems and critical services in terms of improving network resiliency via adding additional nodes and link or rewiring. [ABSTRACT FROM AUTHOR]

This paper provides a survey of recent research progress in mathematical modeling and distributed control of wide-area networks. Firstly, the modeling is introduced for two types of wide-area networks, i.e., coopetitive networks and cooperative networks, with the help of algebraic graph theory. Particularly, bipartite network topologies and cluster network topologies are introduced for coopetitive networks. With respect to cooperative networks, an intermittent clustered network modeling is presented. Then, some classical distributed control strategies are reviewed for wide-area networks to ensure some desired collective behaviors, such as consensus (or synchronization), bipartite consensus (or polarization), and cluster consensus (or fragmentation). Finally, some conclusions and future directions are summarized. [ABSTRACT FROM AUTHOR]

Rare genetic diseases are often caused by single-gene defects that affect various biological processes across different scales. However, it is challenging to identify the causal genes and understand the molecular mechanisms of these diseases. In this paper, we present a multiplex network approach to study the relationship between human diseases and genes. We construct a human disease network (HDN) and a human genome network (HGN) based on genotype–phenotype associations and gene interactions, respectively. We analyze 3771 rare diseases and find distinct phenotypic modules within each dimension that reflect the functional effects of gene mutations. These modules can also be used to predict novel gene candidates for unsolved rare diseases and to explore the cross-scale impact of gene perturbations. We compute various centrality measures for both networks and compare them. Our main finding is that diseases are weakly connected in the HDN, while genes are strongly connected in the HGN. This implies that diseases are relatively isolated from each other, while genes are involved in multiple biological processes. This result has implications for understanding the transmission of infectious diseases and the development of therapeutic interventions. We also show that not all diseases have the same potential to spread infections to other parts of the body, depending on their centrality in the HDN. Our results show that the phenotypic module formalism can capture the complexity of rare diseases beyond simple physical interaction networks and can be applied to study diseases arising from DNA (Deoxyribonucleic Acid) sequencing errors. This study provides a novel network-based framework for integrating multi-scale data and advancing the understanding and diagnosis of rare genetic diseases. [ABSTRACT FROM AUTHOR]

Reducing the number of crossings on graph edges can be useful in various applications, including network visualization, circuit design, graph theory, cartography or social choice theory. This paper aims to determine the crossing number of the join product G * + D n , where G * is a connected graph isomorphic to K 2 , 2 , 2 ∖ { e 1 , e 2 } obtained by removing two edges e 1 , e 2 with a common vertex and a second vertex from the different partitions of the complete tripartite graph K 2 , 2 , 2 , and D n is a discrete graph composed of n isolated vertices. The proofs utilize known exact crossing number values for join products of specific subgraphs H k of G * with discrete graphs in combination with the separating cycles. Similar approaches can potentially estimate unknown crossing numbers of other six-vertex graphs with a larger number of edges in join products with discrete graphs, paths or cycles. [ABSTRACT FROM AUTHOR]

In this paper, the issue of exponential synchronization in Markov switching inertial neural networks with mixed delays is investigated via aperiodically on–off adaptive control. The inertial term is considered, which extends the existing network modes with first-order differential term. Combined with the Lyapunov method, graph theory, and the differential inequalities technique, two types of synchronization criteria are presented which take into account all of the time delay information and reduce the conservativeness. Finally, some numerical simulations are provided in order to show the validity of the theoretical results. [ABSTRACT FROM AUTHOR]

Computing directed Minimum Spanning Tree ( D M S T ) is a fundamental problem in graph theory. It is applied in a wide spectrum of fields from computer network and communication protocol design to revenue maximization in social networks and syntactic parsing in Natural Language Processing. State-of-the-art solutions are online algorithms that compute  D M S T  for a given graph and a root. For multi-query requirements, the online algorithm is inefficient. To overcome the drawbacks, in this paper, we propose an indexed approach that reuses the computation result to facilitate single and batch queries. We store all the potential edges of  D M S T  in a hierarchical tree in  O (n)  space complexity. Furthermore, we answer the  D M S T  query of any root in  O (n)  time complexity. Experimental results demonstrate that our approach can achieve a speedup of 2–3 orders of magnitude in query processing compared to the state-of-the-art while consuming O(n) index size. [ABSTRACT FROM AUTHOR]

Enhancing underwater images presents a challenging problem owing to the influence of ocean currents, the refraction, absorption and scattering of light by suspended particles, and the weak illumination intensity. Recently, different methods have relied on the underwater image formation model and deep learning techniques to restore underwater images. However, they tend to degrade the underwater images, interfere with background clutter and miss the boundary details of blue regions. An improved image fusion and enhancement algorithm based on a prior dark channel is proposed in this paper based on graph theory. Image edge feature sharpening, and dark detail enhancement by homomorphism filtering in CIELab colour space are realized. In the RGB colour space, the multi-scale retinal with colour restoration (MSRCR) algorithm is used to improve colour deviation and enhance colour saturation. The contrast-limited adaptive histogram equalization (CLAHE) algorithm defogs and enhances image contrast. Finally, according to the dark channel images of the three processing results, the final enhanced image is obtained by the linear fusion of multiple images and channels. Experimental results demonstrate the effectiveness and practicality of the proposed method on various data sets. [ABSTRACT FROM AUTHOR]

The explosive increase in educational data and information systems has led to new teaching practices, challenges, and learning processes. To effectively manage and analyze this information, it is crucial to adopt innovative methodologies and techniques. Recommender systems (RSs) offer a solution for advising students and guiding their learning journeys by utilizing statistical methods such as machine learning (ML) and graph analysis to analyze program and student data. This paper introduces an RS for advisors and students that analyzes student records to develop personalized study plans over multiple semesters. The proposed system integrates ideas from graph theory, performance modeling, ML, explainable recommendations, and an intuitive user interface. The system implicitly implements many academic rules through network analysis. Accordingly, a systematic and comprehensive review of different students' plans was possible using metrics developed in the mathematical graph theory. The proposed system systematically assesses and measures the relevance of a particular student's study plan. Experiments on datasets collected at the University of Dubai show that the model presented in this study outperforms similar ML-based solutions in terms of different metrics. Typically, up to 86% accuracy and recall have been achieved. Additionally, the lowest mean square regression (MSR) rate of 0.14 has been attained compared to other state-of-the-art regressors. [ABSTRACT FROM AUTHOR]

The motion of solid objects or even fluids can be described using mathematics. Wind movements, turbulence in the oceans, migration of birds, pandemic of diseases and all other phenomena or systems can be understood using mathematics, i.e., mathematical modelling. Some of the most common techniques used for mathematical modelling are Ordinary Differential Equation (ODE), Partial Differential Equation (PDE), Statistical Methods and Neural Network (NN). However, most of them require substantial amounts of data or an initial governing equation. Furthermore, if a system increases its complexity, namely, if the number and relation between its components increase, then the amount of data required and governing equations increase too. A graph is another well-established concept that is widely used in numerous applications in modelling some phenomena. It seldom requires data and closed form of relations. The advancement in the theory has led to the development of a new concept called autocatalytic set (ACS). In this paper, a new form of ACS, namely, multidigraph autocatalytic set (MACS) is introduced. It offers the freedom to model multi relations between components of a system once needed. The concept has produced some results in the form of theorems and in particular, its relation to the Perron–Frobenius theorem. The MACS Graph Algorithm (MACSGA) is then coded for dynamic modelling purposes. Finally, the MACSGA is implemented on the vector borne disease network system to exhibit MACS's effectiveness and reliability. It successfully identified the two districts that were the main sources of the outbreak based on their reproduction number, R 0 . [ABSTRACT FROM AUTHOR]

Wireless ad hoc networks are the solution for providing network connectivity in challenging environments with a lack or absence of infrastructure. Data transmission in such networks typically adopts geographic routing protocols, which use geographic coordinates as addresses of network devices. However, geographic coordinates are not always obtainable, as traditional localization systems (GNSS, Wi-Fi, terrestrial infrastructure) might not be available due to signal loss. In this paper, we propose a method that assigns virtual coordinates to network nodes, which can be used as input for geographic routing protocols. The numerical results demonstrate the high topological similarity between the physical and the virtual network. Our method exhibits convergence advantages over conventional approaches and outperforms them in terms of the total number of discovered paths. [ABSTRACT FROM AUTHOR]

Complex network science is an interdisciplinary field of study based on graph theory, statistical mechanics, and data science. With the powerful tools now available in complex network theory for the study of network topology, it is obvious that complex network topology models can be applied to enhance artificial neural network models. In this paper, we provide an overview of the most important works published within the past 10 years on the topic of complex network theory-based optimization methods. This review of the most up-to-date optimized neural network systems reveals that the fusion of complex and neural networks improves both accuracy and robustness. By setting out our review findings here, we seek to promote a better understanding of basic concepts and offer a deeper insight into the various research efforts that have led to the use of complex network theory in the optimized neural networks of today. [ABSTRACT FROM AUTHOR]

The Randić index is among the most famous degree-based topological indices in chemical graph theory. It was introduced due to its application in modeling the properties of certain molecular structures and has been extensively studied. In this paper, we study the lower bound of the Randić index of trees in terms of the order and the total domination number. Finally, trees with the minimal Randić index are characterized. [ABSTRACT FROM AUTHOR]

Covert networks refer to criminal organizations that operate outside the boundaries of the law; they can be mainly classified as terrorist networks and criminal networks. We consider how Social Network Analysis (SNA) is used to analyze such networks in order to attain a greater knowledge of criminal behavior. In fact, SNA allows examining the network structure and functioning by computing relevant metrics and parameters to identify roles, positions, features, and other network functioning that are not otherwise easily discovered at first glance. This is why Law Enforcement Agencies (LEAs) are showing growing interest in SNA, which is also used to identify weak spots and disrupt criminal groups. This paper provides a literature review and a classification of methods and real-case applications of disruption techniques. It considers covert network adaptability to such dismantling attempts, herein referred to as resilience. Critical problems of SNA in criminal studies are discussed, including data collection techniques and the inevitable incompleteness and biases of real-world datasets, with the aim of promoting a new research stream for both dismantling techniques and data collection issues. [ABSTRACT FROM AUTHOR]

This paper studies noise-to-state stability in probability (NSSP) for random complex dynamical systems on networks (RCDSN). On the basis of Kirchhoff's matrix theorem in graph theory, an appropriate Lyapunov function which combines with every subsystem for RCDSN is established. Moreover, some sufficient criteria closely related to the topological structure of RCDSN are given to guarantee RCDSN to meet NSSP by means of the Lyapunov method and stochastic analysis techniques. Finally, to show the usefulness and feasibility of theoretical findings, we apply them to random coupled oscillators on networks (RCON), and some numerical tests are given. [ABSTRACT FROM AUTHOR]

The forgotten topological index of a (molecule) graph is the sum of cubes of all its vertex degrees, which plays a significant role in measuring the branching of the carbon atom skeleton. It is meaningful and difficult to explore sufficient conditions for a given graph keeping certain properties in graph theory. In this paper, we mainly explore sufficient conditions in terms of the forgotten topological index for a graph to be ℓ-connected, ℓ-deficient, ℓ-Hamiltonian and ℓ−-independent, respectively. The conditions cannot be dropped. [ABSTRACT FROM AUTHOR]

The use of graph theory for solving labyrinths and mazes is well known, understanding the possible paths as the connections between the nodes that represent the corners or bifurcations. This work presents a new idea: minimizing the length of the optimal path formulated as a topology optimization problem. The maze is mapped with finite elements, and then, through the eigenvalues of the Laplacian matrix of the graph, a constraint is imposed over the connectivity between the input and the output. Several 2D examples are provided to support this approach, allowing for unequivocally finding the shortest path in mazes with multiple connections between entrance and exit, resulting in an nonheuristic algorithm. [ABSTRACT FROM AUTHOR]

The main concept involved in this study explains the theme of complex t-intuitionistic fuzzy graphs (CTIFGs), which act as a powerful tool in analyzing and displaying the relationships among various applications that are difficult to recognize. The manuscript also demonstrates the capability of CTIFGs to create complex associations with multiple domains when considering a physical situation. Following this, the basic set of operations for CTIFGs is projected. The ideas on isomorphism and homomorphism of the CTIFGs are also presented. Moreover, the manuscript describes the importance of the above-mentioned technique in an effective way, giving a solution to the practical application associated with rubber processing industrial wastewater. The contributing factors and corresponding interdependencies are considered when calibrating the complex nature of industrial wastewater associated with the CTIFGs. The results highlight the adaptability and possible efficiencies of CTIFGs, which act as a decision-making tool and also indicate their importance for policy planners in important societal issues. [ABSTRACT FROM AUTHOR]

There are several chemical indices that have been introduced in theoretical chemistry to measure the properties of molecular topology, such as distance-based topological indices, degree-based topological indices and counting-related topological indices. Among the degree-based topological indices, the atom-bond connectivity (ABC) index and geometric-arithmetic (GA) index are the most important, because of their chemical significance. Certain physicochemical properties, such as the boiling point, stability and strain energy, of chemical compounds are correlated by these topological indices. In this paper, we study the molecular topological properties of some special graphs. The indices (ABC), (ABC4), (GA) and (GA5) of these special graphs are computed. [ABSTRACT FROM AUTHOR]

Fuzzy graph theory is a conceptual framework to study and analyze the units that are intensely or frequently connected in a network. It is used to study the mathematical structures of pairwise relations among objects. An m-polar fuzzy (mF, for short) set is a useful notion in practice, which is used by researchers or modelings on real world problems that sometimes involve multi-agents, multi-attributes, multi-objects, multi-indexes and multi-polar information. In this paper, we apply the concept of mF sets to hypergraphs, and present the notions of regular mF hypergraphs and totally regular mF hypergraphs. We describe the certain properties of regular mF hypergraphs and totally regular mF hypergraphs. We discuss the novel applications of mF hypergraphs in decision-making problems. We also develop efficient algorithms to solve decision-making problems. [ABSTRACT FROM AUTHOR]

The main objective of this research is to establish a new formulation and mathematical model based on graph theory to create dynamic equations and provide clarity on the fundamental formulation. We have employed graph theory as a new approach to develop a new representation and formulate the vertical dynamics of a motorcycle with four degrees of freedom, including a suspension and tire model. We have outlined the principal procedural steps required to generate the mathematical and dynamic equations. This systematic approach ensures clarity and precision in our formulation process and representation. Subsequently, we implemented the dynamics equations to examine the dynamic behavior of both the sprung and unsprung masses' vertical displacements, while considering the varying conditions of the road profile. [ABSTRACT FROM AUTHOR]

Feynman diagrams represent one of the most powerful and fascinating tools developed in theoretical physics in the last century. Introduced within the framework of quantum electrodynamics as a suitable method for computing the amplitude of a physical process, they rapidly became a fundamental mathematical object in quantum field theory. However, their abstract nature seems to suggest a wider usage, which actually exceeds the physical context. Indeed, as mathematical objects, they could simply be considered graphs that depict not only physical quantities but also biological or economic entities. We survey the analytical and algebraic properties of such diagrams to understand their utility in several areas of science, eventually providing some examples of recent applications. [ABSTRACT FROM AUTHOR]

The presented paper is devoted to an innovative way of teaching mathematics, specifically the subject combinatorics in high schools. This is because combinatorics is closely connected with the beginnings of informatics and several other scientific disciplines such as graph theory and complexity theory. It is important in solving many practical tasks that require the compilation of an object with certain properties, proves the existence or non-existence of some properties, or specifies the number of objects of certain properties. This paper examines the basic combinatorial structures and presents their use and learning using relations through the Placemat method in teaching process. The effectiveness of the presented innovative way of teaching combinatorics was also verified experimentally at a selected high school in the Slovak Republic. Our experiment has confirmed that teaching combinatorics through relationships among talented children in mathematics is more effective than teaching by a standard algorithmic approach. [ABSTRACT FROM AUTHOR]

A hypergraph is the most developed tool for modeling various practical problems in different fields, including computer sciences, biological sciences, social networks and psychology. Sometimes, given data in a network model are based on bipolar information rather than one sided. To deal with such types of problems, we use mathematical models that are based on bipolar fuzzy (BF) sets. In this research paper, we introduce the concept of BF directed hypergraphs. We describe certain operations on BF directed hypergraphs, including addition, multiplication, vertex-wise multiplication and structural subtraction. We introduce the concept of B = (m+,m-)-tempered BF directed hypergraphs and investigate some of their properties. We also present an algorithm to compute the minimum arc length of a BF directed hyperpath. [ABSTRACT FROM AUTHOR]

Deep learning technology has greatly propelled the development of intelligent and information-driven research on ship infrared automatic target recognition (SIATR). In future scenarios, there will be various recognition models with different mechanisms to choose from. However, in complex and dynamic environments, ship infrared (IR) data exhibit rich feature space distribution, resulting in performance variations among SIATR models, thus preventing the existence of a universally superior model for all recognition scenarios. In light of this, this study proposes a model-matching method for SIATR tasks based on bipartite graph theory. This method establishes evaluation criteria based on recognition accuracy and feature learning credibility, uncovering the underlying connections between IR attributes of ships and candidate models. The objective is to selectively recommend the optimal candidate model for a given sample, enhancing the overall recognition performance and applicability of the model. We separately conducted tests for the optimization of accuracy and credibility on high-fidelity simulation data, achieving Accuracy and EDMS (our credibility metric) of 95.86% and 0.7781. Our method improves by 1.06% and 0.0274 for each metric compared to the best candidate models (six in total). Subsequently, we created a recommendation system that balances two tasks, resulting in improvements of 0.43% (accuracy) and 0.0071 (EDMS). Additionally, considering the relationship between model resources and performance, we achieved a 28.35% reduction in memory usage while realizing enhancements of 0.33% (accuracy) and 0.0045 (EDMS). [ABSTRACT FROM AUTHOR]

A graph with n vertices is called an n-graph. A spanning tree with at most k leaves is referred to as a spanning k-ended tree. Spanning k-ended trees are important in various fields such as network design, graph theory, and communication networks. They provide a structured way to connect all the nodes in a network while ensuring efficient communication and minimizing unnecessary connections. In addition, they serve as fundamental components for algorithms in routing, broadcasting, and spanning tree protocols. However, determining whether a connected graph has a spanning k-ended tree or not is NP-complete. Therefore, it is important to identify sufficient conditions for the existence of such trees. The implicit-degree proposed by Zhu, Li, and Deng is an important indicator for the Hamiltonian problem and the spanning k-ended tree problem. In this article, we provide two sufficient conditions for K1,4-free connected graphs to have spanning k-ended trees for k = 2, 3. We prove the following: Let G be a K1,4-free connected n-graph. For k = 2, 3, if the implicit-degree sum of any k + 1 independent vertices of G is at least n − k + 2, then G has a spanning k-ended tree. Moreover, we give two examples to show that the lower bounds n and n − 1 are the best possible. [ABSTRACT FROM AUTHOR]

Graph theory is a well-established concept that is widely used in numerous applications such as in biology, chemistry and network analysis. The advancement in the theory of graph has led to the development of a new concept called fuzzy autocatalytic set. In this paper, a fuzzy graph-based chemometrics method, namely, chemometrics fuzzy autocatalytic set (c-FACS) is developed and applied for gelatin authentication. The issue on authenticity of gelatin has become a sensitive issue among some religious communities. Due to the matter, Fourier transform infrared (FTIR) spectra of bovine, porcine and fish gelatins are analyzed using c-FACS to identify their signatures and differences and presented in this paper. The results from the c-FACS analysis showed distinct features of each gelatin, particularly porcine. Furthermore, the new method is faster than principal component analysis (PCA) in identifying the gelatin sources. [ABSTRACT FROM AUTHOR]

The concept of smart cities emerged in the 1990s. Since then, smart buildings have become a closely interconnected element of smart cities. This type of building implements Internet of Things technology and control algorithms to monitor and control their indoor environment. The aim of this paper is to develop a new stability criterion method for smart building Internet of Things (IoT) systems, subject to external disturbances. The new stability criterion is going to optimize the operation of control algorithms since this criterion does not depend on the transmission function of the control algorithm but on the data collected by the IoT system. We present a new matrix called "Laplacian IoT matrix", containing IoT network information associated with the graph of a smart building. The proposal is supported by the results of a numerical case study. [ABSTRACT FROM AUTHOR]

Focusing first on the case HT ht , they determine bounds on the number of edges. Finally, the case of a fixed value HT ht is also discussed. They determine the graphs with largest HT ht -spectral radius with fixed vertex or edge connectivity. In addition, the corresponding extremal graphs are given and equations satisfying the HT ht -spectral radius are derived. [Extracted from the article]

Graph theory serves as an engaging arena for the investigation of proof methods within the field of discrete mathematics, and its findings find practical utility in numerous scientific domains. Chemical graph theory is a specialized branch of mathematics that uses graphs to represent and analyze the structure and properties of chemical compounds. Topological indices are mathematical properties of graphs that play a crucial role in chemistry. They provide a unique way to connect the structural characteristics of chemical compounds to their corresponding molecular graphs. The flabellum graph F n (k , j) is obtained with the help of k ≥ 2 duplicates of the cycle graph C n with a common vertex (known as, central vertex). Then, in j of these duplicates, additional edges are added, joining the central vertex to all non-adjacent vertices. In this article, we compute different degree-based topological indices for flabellum graphs, including some well known indices, such as the Randić index, the atom bond connectivity index, the geometric–arithmetic index, and the Zagreb indices. This research provides an in-depth examination of these specific indices within the context of flabellum graphs. Moreover, the behavior of these indices is shown graphically, in terms of the parameters j , k , and n. Additionally, we have extended the concept of the first Zagreb index, to address the issue of cybercrime. This application enables us to identify criminals who exhibit higher levels of activity and engagement in multiple criminal activities when compared to their counterparts. Furthermore, we conducted a comprehensive comparative analysis of the first Zagreb index against the closeness centrality measure. This analysis sheds light on the effectiveness and relevance of the topological index in the context of cybercrime detection and network analysis. [ABSTRACT FROM AUTHOR]

It is well known that crisp graph theory is saturated. However, saturation in a fuzzy environment has only lately been created and extensively researched. It is necessary to consider m components for each node and edge in an m-polar fuzzy graph. Since there is only one component for this idea, we are unable to manage this kind of circumstance using the fuzzy model since we take into account m components for each node as well as edges. Again, since each edge or node only has two components, we are unable to apply a bipolar or intuitionistic fuzzy graph model. In contrast to other fuzzy models, mPFG models produce outcomes of fuzziness that are more effective. Additionally, we develop and analyze these kinds of mPFGs using examples and related theorems. Considering all those things together, we define saturation for a m-polar fuzzy graph (mPFG) with multiple membership values for both vertices and edges; thus, a novel approach is required. In this context, we present a novel method for defining saturation in mPFG involving m saturations for each element in the membership value array of a vertex. This explains α -saturation and β -saturation. We investigate intriguing properties such as α -vertex count and β -vertex count and establish upper bounds for particular instances of mPFGs. Using the concept of α -saturation and α -saturation, block and bridge of mPFG are characterized. To identify the α -saturation and β -saturation mPFGs, two algorithms are designed and, using these algorithms, the saturated mPFG is determined. The time complexity of these algorithms is O (| V | 3) , where | V | is the number of vertices of the given graph. In addition, we demonstrate a practical application where the concept of saturation in mPFG is applicable. In this application, an appropriate location is determined for the allocation of a facility point. [ABSTRACT FROM AUTHOR]

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 Z n , 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. [ABSTRACT FROM AUTHOR]

The study on graphs emerging from different algebraic structures such as groups, rings, fields, vector spaces, etc. is a prominent area of research in mathematics, as algebra and graph theory are two mathematical fields that focus on creating and analysing structures. There are numerous studies linking algebraic structures and graphs, which began with the introduction of Cayley graphs of groups. Several algebraic graphs have been defined on rings, a fast-growing area in the literature. In this article, we systematically review the literature on some variants of Cayley graphs that are defined on rings and highlight the properties and characteristics of such graphs, to showcase the research in this area. [ABSTRACT FROM AUTHOR]

This paper deals with some theoretical aspects of hypergraphs related to hyperpaths and hypertrees. In ordinary graph theory, the intersecting or adjacent edges contain exactly one vertex; however, in the case of hypergraph theory, the adjacent or intersecting hyperedges may contain more than one vertex. This fact leads to the intuitive notion of knots, i.e., a collection of explicit vertices. The key idea of this manuscript lies in the introduction of the concept of the knot, which is a subset of the intersection of some intersecting hyperedges. We define knot-hyperpaths and equivalent knot-hyperpaths and study their relationships with the algebraic space continuity and the pseudo-open character of maps. Moreover, we establish a sufficient condition under which a hypergraph is a hypertree, without using the concept of the host graph. [ABSTRACT FROM AUTHOR]

The energy E (G) of a graph G is the sum of the absolute values of its adjacency matrix. In contrast, the trace norm of a digraph Q, which is the sum of the singular values of the corresponding adjacency matrix, is the oriented version of the energy of a graph. It is worth pointing out that one of the main problems in this theory consists of determining appropriated bounds of these types of energies for significant classes of graphs, digraphs and matrices, provided that, in general, finding out their exact values is a problem of great difficulty. In this paper, the trace norm of a { 0 , 1 } -Brauer configuration is introduced. It is estimated and computed by associating suitable families of graphs and posets to Brauer configuration algebras. [ABSTRACT FROM AUTHOR]

Based on the graph theory and stability theory of dynamical system, this paper studies the stability of the trivial solution of a coupled fractional-order system. Some sufficient conditions are obtained to guarantee the global stability of the trivial solution. Finally, a comparison between fractional-order system and integer-order system ends the paper. [ABSTRACT FROM AUTHOR]

Fuzzy graph theory is a useful and well-known tool to model and solve many real-life optimization problems. Since real-life problems are often uncertain due to inconsistent and indeterminate information, it is very hard for an expert to model those problems using a fuzzy graph. A neutrosophic graph can deal with the uncertainty associated with the inconsistent and indeterminate information of any real-world problem, where fuzzy graphs may fail to reveal satisfactory results. The concepts of the regularity and degree of a node play a significant role in both the theory and application of graph theory in the neutrosophic environment. In this work, we describe the utility of the regular neutrosophic graph and bipartite neutrosophic graph to model an assignment problem, a road transport network, and a social network. For this purpose, we introduce the definitions of the regular neutrosophic graph, star neutrosophic graph, regular complete neutrosophic graph, complete bipartite neutrosophic graph, and regular strong neutrosophic graph. We define the d m - and t d m -degrees of a node in a regular neutrosophic graph. Depending on the degree of the node, this paper classifies the regularity of a neutrosophic graph into three types, namely d m -regular, t d m -regular, and m-highly irregular neutrosophic graphs. We present some theorems and properties of those regular neutrosophic graphs. The concept of an m-highly irregular neutrosophic graph on cycle and path graphs is also investigated in this paper. The definition of busy and free nodes in a regular neutrosophic graph is presented here. We introduce the idea of the μ -complement and h-morphism of a regular neutrosophic graph. Some properties of complement and isomorphic regular neutrosophic graphs are presented here. [ABSTRACT FROM AUTHOR]

In this paper, we consider fault-tolerant resolving sets in graphs. We characterize n-vertex graphs with fault-tolerant metric dimension n, n − 1 , and 2, which are the lower and upper extremal cases. Furthermore, in the first part of the paper, a method is presented to locate fault-tolerant resolving sets by using classical resolving sets in graphs. The second part of the paper applies the proposed method to three infinite families of regular graphs and locates certain fault-tolerant resolving sets. By accumulating the obtained results with some known results in the literature, we present certain lower and upper bounds on the fault-tolerant metric dimension of these families of graphs. As a byproduct, it is shown that these families of graphs preserve a constant fault-tolerant resolvability structure. [ABSTRACT FROM AUTHOR]

In view of the wide application of resistance distance, the computation of resistance distance in various graphs becomes one of the main topics. In this paper, we aim to compute resistance distance in H-join of graphs G 1 , G 2 , ... , G k . Recall that H is an arbitrary graph with V (H) = { 1 , 2 , ... , k } , and G 1 , G 2 , ... , G k are disjoint graphs. Then, the H-join of graphs G 1 , G 2 , ... , G k , denoted by ⋁ H { G 1 , G 2 , ... , G k } , is a graph formed by taking G 1 , G 2 , ... , G k and joining every vertex of G i to every vertex of G j whenever i is adjacent to j in H. Here, we first give the Laplacian matrix of ⋁ H { G 1 , G 2 , ... , G k } , and then give a { 1 } -inverse L (⋁ H { G 1 , G 2 , ... , G k }) { 1 } or group inverse L (⋁ H { G 1 , G 2 , ... , G k }) # of L (⋁ H { G 1 , G 2 , ... , G k }) . It is well know that, there exists a relationship between resistance distance and entries of { 1 } -inverse or group inverse. Therefore, we can easily obtain resistance distance in ⋁ H { G 1 , G 2 , ... , G k } . In addition, some applications are presented in this paper. [ABSTRACT FROM AUTHOR]