Search

We extend the notion of balance from the realm of signed and gain graphs to conjugate skew gain graphs which are skew gain graphs where the labels on the oriented edges get conjugated when we reverse the orientation. We characterize the balance in a conjugate skew gain graph in several ways especially by dealing with its adjacency matrix and the g -Laplacian matrix. We also deal with the concept of anti-balance in conjugate skew gain graphs. [ABSTRACT FROM AUTHOR]

Graph theory is a branch of mathematics that is crucial to modelling applicable systems and networks using matrix representations. In this article, a novel graph-theoretic model was used to assess an urban water distribution system (WDS) in Saudi Arabia. This graph model is based on representing its elements through nodes and links using a weighted adjacency matrix. The nodes represent the points where there can be a water input or output (sources, treatment plants, tanks, reservoirs, consumers, connections), and links represent the edges of the graph that carry water from one node to another (pipes, pumps, valves). Four WDS benchmarks, pumps, tanks, reservoirs, and external sources were used to validate the framework at first. This validation showed that the worst-case scenarios for vulnerability were provided by the fault sequence iterating the calculation of the centrality measurements. The vulnerability framework’s application to the Saudi Arabian WDS enabled the identification of the system’s most vulnerable junctions and zones. As anticipated, the regions with the fewest reservoirs were most at risk from unmet demand, indicating that this system is vulnerable to the removal of junctions and pipes that are intricately associated with their neighbours. Different centrality metrics were computed, from which the betweenness centrality offered the worst vulnerability prediction measures. The aspects and zones of the WDS that can more significantly impact the water supply in the event of a failure were identified by the vulnerability framework utilising attack tactics.

We explore the relationship between lattice field theory and graph theory, placing special emphasis on the interplay between Dirac and scalar lattice operators and matrices within the realm of spectral graph theory. Beyond delving into fundamental concepts of spectral graph theory, such as adjacency and Laplacian matrices, we introduce a novel matrix called an "antisymmetrized adjacency matrix", specifically tailored for cycle digraphs (T 1 lattice) and simple directed paths (B 1 lattice). The nontrivial relationship between graph theory matrices and lattice operators shows that the graph Laplacian matrix mirrors the lattice scalar operator and the Wilson term in lattice fermions, while the antisymmetrized adjacency matrix, along with its extensions to higher dimensions, is equivalent to naive lattice Dirac operators. Building upon these connections, we provide rigorous proofs for two key assertions: (i) The count of zero-modes in a free lattice scalar operator coincides with the zeroth Betti number of the underlying graph (lattice). (ii) The maximum count of Dirac zero-modes in a free lattice fermion operator is equivalent to the cumulative sum of all Betti numbers when the D -dimensional graph results from a Cartesian product of cycle digraphs (T 1 lattice) and simple directed paths (B 1 lattice). [ABSTRACT FROM AUTHOR]

Let G be a graph with the adjacency matrix A (G) , and let D (G) be the diagonal matrix of the degrees of G. Nikiforov first defined the matrix A α (G) as A α (G) = α D (G) + (1 − α) A (G) , 0 ≤ α ≤ 1 , which shed new light on A (G) and Q (G) = D (G) + A (G) , and yielded some surprises. The α − adjacency energy E A α (G) of G is a new invariant that is calculated from the eigenvalues of A α (G) . In this work, by combining matrix theory and the graph structure properties, we provide some upper and lower bounds for E A α (G) in terms of graph parameters (the order n, the edge size m, etc.) and characterize the corresponding extremal graphs. In addition, we obtain some relations between E A α (G) and other energies such as the energy E (G) . Some results can be applied to appropriately estimate the α -adjacency energy using some given graph parameters rather than by performing some tedious calculations. [ABSTRACT FROM AUTHOR]

If A(G) and D(G) are respectively the adjacency matrix and the diagonal matrix of vertex degrees of a connected graph G, the generalized adjacency matrix Aα(G) is defined as Aα(G) = α D(G) + (1 - α) A(G), where 0 ≤ α ≤ 1. The Aα (or generalized) spectral radius λ(Aα(G)) (or simply λα) is the largest eigenvalue of Aα(G). In this paper, we show that λα ≤ α Δ + (1 - α)√ 2m ( 1 - 1/ω), where m, Δ and ω = ω(G) are respectively the size, the largest degree and the clique number of G. Further, if G has order n, then we show that ... where di and mi are respectively the degree and the average 2-degree of the vertex Vi. [ABSTRACT FROM AUTHOR]

A Modular Reconfigurable Robot (MRR) composed of standard joint and link modules has the advantages of rapid changeover of assembly configurations for a diversity of application requirements. To tackle the kinematics analysis issues for an MRR, it is essential to develop a generic method to mathematically represent all possible assembly configurations and automatically generate their kinematic models. In this paper, two types of robot modules, i.e., revolute joint modules and rigid link modules, are considered as the basic building blocks of an MRR. The topological structure of an MRR is represented by a graph, termed an Assembly Graph (AG), in which all robot modules, regardless of their types, are treated as vertices, while the connecting interfaces between any two robot modules are treated as edges. Based on graph theory, a modified adjacency matrix, termed an Assembly Adjacency Matrix (AAM), is proposed to represent the assembly configuration of an MRR, in which the three-dimensional assembly information is specified with connecting port vectors for the adjacent modules. A path matrix derived from the AAM of an MRR is employed to describe the connecting sequence of its constituting modules. The local frame representation of the Product-of-Exponentials (POE) formula is employed, which is configuration-independent. Therefore, for an MRR configuration represented with an AAM, its kinematic model will be automatically generated according to its path matrix. The proposed assembly configuration representation and kinematic modeling method are validated through a dual-branch MRR configuration. [ABSTRACT FROM AUTHOR]

The traditional modelling and simulation analysis methods based on graph theory are difficult to present the structural characteristics and dynamic laws of cyber-physical power systems (CPPS). In order to reduce the complexity of the attack graph, this paper proposes a CPPS robustness analysis method based on the reachable matrix. Based on the evolution of the augmented adjacency matrix of cyber-physical power system, this method uses the reachable matrix to obtain the maximum connectivity matrix of the system (the adjacency matrix corresponding to the largest connected subgraph of cyber-physical power system), and then, evaluates and analyses the robustness of cyber-physical power system. The IEEE118 node standard model is taken as an example to carry out the simulation experiment. The experimental results show that the reachable matrix has feasibility in the robustness analysis of cyber-physical power systems. Further through the contrast experiment found that compared with the robustness analysis method based on graph theory, the proposed method in this paper can reduce the modelling complexity of the mutual coupling network, reduce the complexity of the attack graph, and clearly and intuitively show the structural characteristics and dynamic laws of the network connection in the system. [ABSTRACT FROM AUTHOR]

For the adjacency matrix A(G) and diagonal matrix of the vertex degrees D(G) of a simple graph G, the A(G) matrix is the convex combinations of D(G) and A(G), and is defined as A(G) = D(G)+(1)A(G), for 0 n be the eigenvalues of A(G) (which we call -adjacency eigenvalues of the graph G). The generalized adjacency energy also called -adjacency energy of the graph G is defined as EA (G) = is the average vertex degree, m is the size and n is the order of G. The -adjacency energy of a graph G merges the theory of energy (adjacency energy) and the signless Laplacian energy, as EA0 (G) = E (G) and 2E A 12 (G) = QE(G), where E (G) is the energy and QE(G) is the signless Laplacian energy of G. In this paper, we obtain some new upper and lower bounds for the generalized adjacency energy of a graph, in terms of different graph parameters like the vertex covering number, the Zagreb index, the number of edges, the number of vertices, etc. We characterize the extremal graphs attained these bounds. [ABSTRACT FROM AUTHOR]

Graph theory is one of the neglected branches of mathematics in Nigeria but with the most applications in other fields of research. This article shows the paucity, importance, and necessity of graph theory in the development of Nigeria. The adjacency matrix and dual graph of the Nigeria map were presented. The graph spectrum and energies (graph energy and Laplacian energy) of the dual graph were computed. Then the chromatic number, maximum degree, minimum spanning tree, graph radius, and diameter, the Eulerian circuit and Hamiltonian paths from the dual graph were obtained and discussed. [ABSTRACT FROM AUTHOR]

The nullity of a graph G is the multiplicity of zero as an eigenvalue of its adjacency matrix. An assignment of weights to the vertices of a graph, that satisfies a zero sum condition over the neighbors of each vertex, and uses maximum number of independent variables is denoted by a high zero sum weighting of the graph. This applicable tool is used to determine the nullity of the graph. Two types of graphs are defined, and the change of their nullities is studied, namely, the graph G+ab constructed from G by adding a new vertex ab which is joint to all neighbors of both vertices a and b of G, and G•ab which is obtained from G+ab by removing both vertices a and b. [ABSTRACT FROM AUTHOR]

This paper is devoted to the study of the double domination in vague graphs, and it is a contribution to the Special Issue "Advances in graph theory and Symmetry/Asymmetry" of Symmetry. Symmetry is one of the most important criteria that illustrate the structure and properties of fuzzy graphs. It has many applications in dominating sets and helps find a suitable place for construction. Vague graphs (VGs), which are a family of fuzzy graphs (FGs), are a well-organized and useful tool for capturing and resolving a range of real-world scenarios involving ambiguous data. In the graph theory, a dominating set (DS) for a graph G * = (X , E) is a subset D of the vertices X so that every vertex which is not in D is adjacent to at least one member of D. The subject of energy in graph theory is one of the most attractive topics serving a very important role in biological and chemical sciences. Hence, in this work, we express the notion of energy on a dominating vague graph (DVG) and also use the concept of energy in modeling problems related to DVGs. Moreover, we introduce a new notion of a double dominating vague graph (DDVG) and provide some examples to explain various concepts introduced. Finally, we present an application of energy on DVGs. [ABSTRACT FROM AUTHOR]

Spectral graph theory is like a special tool for understanding graphs. It helps us find patterns and connections in complex networks, using the magic of eigenvalues. Let G be the graph and A (G) be its adjacency matrix, then G is singular if the determinant of the adjacency matrix A (G) is 0, otherwise it is nonsingular. Within the realm of nonsingular graphs, there is the concept of property R , where each eigenvalue's reciprocal is also an eigenvalue of G. By introducing multiplicity constraints on both eigenvalues and their reciprocals, it becomes property SR. Similarly, the world of nonsingular graphs reveals property − R , where the negative reciprocal of each eigenvalue also finds a place within the spectrum of G. Moreover, when the multiplicity of each eigenvalue and its negative reciprocal is equal, this results in a graph with a property of − SR . Some classes of unweighted nonbipartite graphs are already constructed in the literature with the help of the complete graph K n and a copy of the path graph P 4 satisfying property R but not SR. This article takes this a step further. The main aim is to construct several weighted classes of graphs which satisfy property R but not SR. For this purpose, the weight functions are determined that enable these nonbipartite graph classes to satisfy the − SR and R properties, even if the unweighted graph does not satisfy these properties. Some examples are presented to support the investigated results. These examples explain how certain weight functions make these special types of graphs meet the properties R or − SR , even when the original graphs without weights do not meet these properties. [ABSTRACT FROM AUTHOR]

In skeleton‐based action recognition, the graph convolutional network (GCN) has achieved great success. Modelling skeleton data in a suitable spatial‐temporal way and designing the adjacency matrix are crucial aspects for GCN‐based methods to capture joint relationships. In this study, we propose the spatial‐temporal slowfast graph convolutional network (STSF‐GCN) and design the adjacency matrices for the skeleton data graphs in STSF‐GCN. STSF‐GCN contains two pathways: (1) the fast pathway is in a high frame rate, and joints of adjacent frames are unified to build 'small' spatial‐temporal graphs. A new spatial‐temporal adjacency matrix is proposed for these 'small' spatial‐temporal graphs. Ablation studies verify the effectiveness of the proposed adjacency matrix. (2) The slow pathway is in a low frame rate, and joints from all frames are unified to build one 'big' spatial‐temporal graph. The adjacency matrix for the 'big' spatial‐temporal graph is obtained by computing self‐attention coefficients of each joint. Finally, outputs from two pathways are fused to predict the action category. STSF‐GCN can efficiently capture both long‐range and short‐range spatial‐temporal joint relationships. On three datasets for skeleton‐based action recognition, STSF‐GCN can achieve state‐of‐the‐art performance with much less computational cost. [ABSTRACT FROM AUTHOR]

Graph theory can be used to address problems with complex network structures. Congenital heart diseases (CHDs) involve complex abnormal connections between chambers, vessels, and organs. We proposed a new method to represent CHDs based on graph theory, wherein vertices were defined as the spaces through which blood flows and edges were defined by the blood flow between the spaces and direction of the blood flow. The CHDs of tetralogy of Fallot (TOF) and transposition of the great arteries (TGA) were selected as examples for constructing directed graphs and binary adjacency matrices. Patients with totally repaired TOF, surgically corrected d-TGA, and Fontan circulation undergoing four-dimensional (4D) flow magnetic resonance imaging (MRI) were included as examples for constructing the weighted adjacency matrices. The directed graphs and binary adjacency matrices of the normal heart, extreme TOF undergoing a right modified Blalock–Taussig shunt, and d-TGA with a ventricular septal defect were constructed. The weighted adjacency matrix of totally repaired TOF was constructed using the peak velocities obtained from 4D flow MRI. The developed method is promising for representing CHDs and may be helpful in developing artificial intelligence and conducting future research on CHD. [ABSTRACT FROM AUTHOR]

One of the most important concepts in graph theory for dealing with unpredictable phenomena is the concept of domination and it has gained attention from many scholars. Recently, dominating energy of graph plays a vital role in the field of graph energies. If the fuzzy graph (FG) fails to give outstanding results, the neutrosophic set (NS) and neutrosophic graphs (NG) can handle the uncertainty correlated with indeterminate and inconsistent information in any real-world scenario. Recent studies related to domination energy in fuzzy environment only deal with the single membership function. It is more flexible and applicable to use bipolar neutrosophic models because they include both positive and negative influencers. Therefore, this paper is based on some developments of neutrosophic graph theory to deal with situations where imprecision is characterized by positive and negative types of membership functions. A novel concept of the dominating energy graph is proposed based on recently introduced concept of bipolar single-valued neutrosophic graphs (BSVNG). Moreover, this study analyses the concepts of dominating energy graph in BSVNG environment. More precisely, the adjacency matrix of a dominating BSVNG as well as the spectrum of the adjacency matrix and their related theory is developed with the help of illustrative examples. Further, the domination energy of BSVNG is computed. Aside from it, various operations relating to this dominating have been depicted. The complement, union, and join of dominating energy in BSVNG have been investigated by using appropriate examples and some properties of the dominating energy in BSVNG are established. [ABSTRACT FROM AUTHOR]

Graph theory is a branch of mathematics that is crucial to modelling applicable systems and networks using matrix representations. In this article, a novel graph-theoretic model was used to assess an urban water distribution system (WDS) in Saudi Arabia. This graph model is based on representing its elements through nodes and links using a weighted adjacency matrix. The nodes represent the points where there can be a water input or output (sources, treatment plants, tanks, reservoirs, consumers, connections), and links represent the edges of the graph that carry water from one node to another (pipes, pumps, valves). Four WDS benchmarks, pumps, tanks, reservoirs, and external sources were used to validate the framework at first. This validation showed that the worst-case scenarios for vulnerability were provided by the fault sequence iterating the calculation of the centrality measurements. The vulnerability framework's application to the Saudi Arabian WDS enabled the identification of the system's most vulnerable junctions and zones. As anticipated, the regions with the fewest reservoirs were most at risk from unmet demand, indicating that this system is vulnerable to the removal of junctions and pipes that are intricately associated with their neighbours. Different centrality metrics were computed, from which the betweenness centrality offered the worst vulnerability prediction measures. The aspects and zones of the WDS that can more significantly impact the water supply in the event of a failure were identified by the vulnerability framework utilising attack tactics. [ABSTRACT FROM AUTHOR]

The quinary reciprocal system Li+,K+||F–,Cl–,Br–, has been divided into simplexes using graph theory: by making up an adjacency matrix and solving a logical expression. The results have been used to construct the phase tree of the system, which has a linear structure and consists of four stable secant tetrahedra, four stable pentatopes, and one stable hexatope. The number and composition of phases crystallizing in the system have been predicted. The LiF–KCl–KBr–Li2CrO4 stable tetrahedron has been studied using differential thermal analysis and X-ray diffraction. The tetrahedron has no invariant equilibrium points. The continuous series of solid solutions between potassium chloride and potassium bromide has been shown to be stable, without decomposition. Three solid phases have been shown to crystallize in the tetrahedron: LiF, Li2CrO4, and KClxBr1–x. [ABSTRACT FROM AUTHOR]

We establish a list of characterizations of bounded twin-width for hereditary classes of totally ordered graphs: as classes of at most exponential growth studied in enumerative combinatorics, as monadically NIP classes studied in model theory, as classes that do not transduce the class of all graphs studied in finite model theory, and as classes for which model checking first-order logic is fixed-parameter tractable studied in algorithmic graph theory. This has several consequences. First, it allows us to show that every hereditary class of ordered graphs either has at most exponential growth, or has at least factorial growth. This settles a question first asked by Balogh et al. [5] on the growth of hereditary classes of ordered graphs, generalizing the Stanley-Wilf conjecture/Marcus-Tardos theorem. Second, it gives a fixed-parameter approximation algorithm for twin-width on ordered graphs. Third, it yields a full classification of fixed-parameter tractable first-order model checking on hereditary classes of ordered binary structures. Fourth, it provides a model-theoretic characterization of classes with bounded twin-width. Finally, it settles the small conjecture [8] in the case of ordered graphs. [ABSTRACT FROM AUTHOR]

This article documents enu, a freely-downloadable, open-source and stand-alone program written in C++ for the enumeration of the constitutional isomers and stereoisomers of a molecular formula. The program relies on graph theory to enumerate all the constitutional isomers of a given formula on the basis of their canonical adjacency matrix. The stereoisomers of a given constitutional isomer are enumerated as well, on the basis of the automorphism group of this matrix. The isomer list is then reported in the form of canonical SMILES strings within files in XML format. The specification of the molecule family of interest is very flexible and the code is optimized for computational efficiency. The algorithms and implementations underlying enu are described, and simple illustrative applications are presented. The enu code is freely available on GitHub at https://github.com/csms-ethz/CombiFF. [ABSTRACT FROM AUTHOR]

The index ≥1(1) of a signed graph 1= (G; -) is just the largest eigenvalue of its adjacency matrix. For any n > 4 we identify the signed graphs achieving the minimum index in the class of signed bicyclic graphs with n vertices. Apart from the n = 4 case, such graphs are obtained by considering a starlike tree with four branches of suitable length (i.e. four distinct paths joined at their end vertex u) with two additional negative independent edges pairwise joining the four vertices adjacent to u. As a by-product, all signed bicyclic graphs containing a theta-graph and whose index is less than 2 are detected. [ABSTRACT FROM AUTHOR]

A graph containing some edges with probability measures and other edges with uncertain measures is referred to as an uncertain random graph. Numerous real-world problems in social networks and transportation networks can be boiled down to optimization problems in uncertain random graphs. Actually, information in optimization problems in uncertain random graphs is always asymmetric. Regularization is a common optimization problem in graph theory, and the regularity index is a fundamentally measurable indicator of graphs. Therefore, this paper investigates the regularity index of an uncertain random graph within the framework of chance theory and information asymmetry theory. The concepts of k-regularity index and regularity index of the uncertain random graph are first presented on the basis of the chance theory. Then, in order to compute the k-regularity index and the regularity index of the uncertain random graph, a simple and straightforward calculating approach is presented and discussed. Furthermore, we discuss the relationship between the regularity index and the k-regularity index of the uncertain random graph. Additionally, an adjacency matrix-based algorithm that can compute the k-regularity index of the uncertain random graph is provided. Some specific examples are given to illustrate the proposed method and algorithm. Finally, we conclude by highlighting some potential applications of uncertain random graphs in social networks and transportation networks, as well as the future vision of its combination with symmetry. [ABSTRACT FROM AUTHOR]

Openness to experience is one of the big five traits of personality which recently has been the subject of several studies in neuroscience due to its importance in understanding various cognitive functions. However, the neural basis of openness to experience is still unclear. Previous studies have found largely heterogeneous results, suggesting that various brain regions may be involved in openness to experience. Here we suggested that performing structural connectome analysis may shed light on the neural underpinnings of openness to experience as it provides a more comprehensive look at the brain regions that are involved in this trait. Hence, we investigated the involvement of brain network structural features in openness to experience which has not yet been explored to date. The magnetic resonance imaging (MRI) data along with the openness to experience trait score from the self-reported NEO Five-Factor Inventory of 100 healthy subjects were evaluated from Human Connectome Project (HCP). CSD-based whole-brain probabilistic tractography was performed using diffusion-weighted images as well as segmented T1-weighted images to create an adjacency matrix for each subject. Using graph theoretical analysis, we computed global efficiency (GE) and clustering coefficient (CC) which are measures of two important aspects of network organization in the brain: functional integration and functional segregation respectively. Results revealed a significant negative correlation between GE and openness to experience which means that the higher capacity of the brain in combining information from different regions may be related to lower openness to experience. [ABSTRACT FROM AUTHOR]

Despite the huge advances in digital communications in the last decade, physical documents are still the most common media for information transfer, especially in the official context. However, the readily available document processing devices and techniques (printers, scanners, etc.) facilitate the illegal manipulation or imitation of original documents by forgers. Therefore, verification of the authenticity and detection of forgery is of paramount importance to all agencies receiving printed documents. We suggest an unsupervised forgery detection framework that can distinguish whether a document is forged based on the spectroscopy of the document's ink. The spectra of the tested documents inks (original and questioned) were obtained using laser-induced breakdown spectroscopy (LIBS) technology. Then, a correlation matrix of the spectra was calculated for both the original and questioned documents together, which were then transformed into an adjacency matrix aiming at converting it into a weighted network under the concept of graph theory. Clustering algorithms were then applied to the network to determine the number of clusters. The proposed approach was tested under a variety of scenarios and different types of printers (e.g., inkjet, laser, and photocopiers) as well as different kinds of papers. The findings show that the proposed approach provided a high rate of accuracy in identifying forged documents and a high detection speed. It also provides a visual output that is easily interpretable to the non-expert, which provides great flexibility for real-world application. [ABSTRACT FROM AUTHOR]

The index of a signed graph is the largest eigenvalue of its adjacency matrix. Let Un,k,4 be the set of all signed complete graphs of order n whose negative edges induce a unicyclic graph of order k and girth at least 4. In this paper, we identify the signed graphs achieving the maximum index in the class Un,k,4. [ABSTRACT FROM AUTHOR]

Recently, hetero-functional graph theory (HFGT) has developed as a means to mathematically model the structure of large-scale complex flexible engineering systems. It does so by fusing concepts from network science and model-based systems engineering (MBSE). For the former, it utilizes multiple graph-based data structures to support a matrix-based quantitative analysis. For the latter, HFGT inherits the heterogeneity of conceptual and ontological constructs found in model-based systems engineering including system form, system function, and system concept. These diverse conceptual constructs indicate multi-dimensional rather than two-dimensional relationships. This paper provides the first tensor-based treatment of hetero-functional graph theory. In particular, it addresses the "system concept" and the hetero-functional adjacency matrix from the perspective of tensors and introduces the hetero-functional incidence tensor as a new data structure. The tensor-based formulation described in this work makes a stronger tie between HFGT and its ontological foundations in MBSE. Finally, the tensor-based formulation facilitates several analytical results that provide an understanding of the relationships between HFGT and multi-layer networks. [ABSTRACT FROM AUTHOR]

Vague graphs (VGs), belonging to the fuzzy graphs (FGs) family, have good capabilities when faced with problems that cannot be expressed by FGs. The notion of a VG is a new mathematical attitude to model the ambiguity and uncertainty in decision-making issues. A vague fuzzy graph structure (VFGS) is the generalization of the VG. It is a powerful and useful tool to find the influential person in various relations. VFGSs can deal with the uncertainty associated with the inconsistent and indeterminate information of any real-world problems where fuzzy graphs may fail to reveal satisfactory results. Moreover, VGSs are very useful tools for the study of different domains of computer science such as networking, social systems, and other issues such as bioscience and medical science. The subject of energy in graph theory is one of the most attractive topics that is very important in biological and chemical sciences. Hence, in this work, we extend the notion of energy of a VG to the energy of a VFGS and also use the concept of energy in modeling problems related to VFGS. Actually, our purpose is to develop a notion of VFGS and investigate energy and Laplacian energy (LE) on this graph. We define the adjacency matrix (AM) concept, energy, and LE of a VFGS. Finally, we present three applications of the energy in decision-making problems. [ABSTRACT FROM AUTHOR]

For a finite group G and a subset X≠ 0 of G, the commuting graph, indicated by G = C(G, X), is the simple connected graph with vertex set X and two distinct vertices x and y are edge connected in G if and only if they commute in X. The Aα matrix of G is specified as Aα(G) = αD(G) + (1-α)A(G), α ∈ [0, 1], where D(G) is the diagonal matrix of vertex degrees while A(G) is the adjacency matrix of G. In this article, we investigate the Aα matrix for commuting graphs of finite groups and we also find the Aα eigenvalues of the dihedral, the semidihedral and the dicyclic groups. We determine the upper bounds for the largest Aα eigenvalue for these graphs. Consequently, we get the adjacency eigenvalues, the Laplacian eigenvalues, and the signless Laplacian eigenvalues of these graphs for particular values of α. Further, we show that these graphs are Laplacian integral. [ABSTRACT FROM AUTHOR]

The analysis of large, multisite neuroimaging datasets provides a promising means for robust characterization of brain networks that can reduce false positives and improve reproducibility. However, the use of different MRI scanners introduces variability to the data. Managing those sources of variability is increasingly important for the generation of accurate group-level inferences. ComBat is one of the most promising tools for multisite (multiscanner) harmonization of structural neuroimaging data, but no study has examined its application to graph theory metrics derived from the structural brain connectome. The present work evaluates the use of ComBat for multisite harmonization in the context of structural network analysis of diffusion-weighted scans from the Advancing Concussion Assessment in Pediatrics (A-CAP) study. Scans were acquired on six different scanners from 484 children aged 8.00–16.99 years [Mean = 12.37 ± 2.34 years; 289 (59.7%) Male] ~10 days following mild traumatic brain injury (n = 313) or orthopedic injury (n = 171). Whole brain deterministic diffusion tensor tractography was conducted and used to construct a 90 x 90 weighted (average fractional anisotropy) adjacency matrix for each scan. ComBat harmonization was applied separately at one of two different stages during data processing, either on the (i) weighted adjacency matrices (matrix harmonization) or (ii) global network metrics derived using unharmonized weighted adjacency matrices (parameter harmonization). Global network metrics based on unharmonized adjacency matrices and each harmonization approach were derived. Robust scanner effects were found for unharmonized metrics. Some scanner effects remained significant for matrix harmonized metrics, but effect sizes were less robust. Parameter harmonized metrics did not differ by scanner. Intraclass correlations (ICC) indicated good to excellent within-scanner consistency between metrics calculated before and after both harmonization approaches. Age correlated with unharmonized network metrics, but was more strongly correlated with network metrics based on both harmonization approaches. Parameter harmonization successfully controlled for scanner variability while preserving network topology and connectivity weights, indicating that harmonization of global network parameters based on unharmonized adjacency matrices may provide optimal results. The current work supports the use of ComBat for removing multiscanner effects on global network topology. [ABSTRACT FROM AUTHOR]

In text generation, a large language model (LM) makes a choice of each new word based only on the former selection of its context using the softmax function. Nevertheless, the link statistics information of concurrent words based on a scene-specific corpus is valuable in choosing the next word, which can help to ensure the topic of the generated text to be aligned with the current task. To fully explore the co-occurrence information, we propose a graphmax function for task-specific text generation. Using the graph-based regularization, graphmax enables the final word choice to be determined by both the global knowledge from the LM and the local knowledge from the scene-specific corpus. The traditional softmax function is regularized with a graph total variation (GTV) term, which incorporates the local knowledge into the LM and encourages the model to consider the statistical relationships between words in a scene-specific corpus. The proposed graphmax is versatile and can be readily plugged into any large pre-trained LM for text generation and machine translation. Through extensive experiments, we demonstrate that the new GTV-based regularization can improve performances in various natural language processing (NLP) tasks in comparison with existing methods. Moreover, through human experiments, we observe that participants can easily distinguish the text generated by graphmax or softmax. [ABSTRACT FROM AUTHOR]

TEMO (topological effect on molecular orbitals) problem is an interesting topic in chemical graph theory. In 2022, Gutman studied TEMOtype regularity for the degree-based topological indices. In this paper, we show that the general TEMO-type regularity holds for the (a, b)-KA index, the Aa-spectral radius and the third leap Zagreb index. [ABSTRACT FROM AUTHOR]

Perinatal stroke occurs early in life and often leads to a permanent, disabling weakness to one side of the body. To test the hypothesis that non-lesioned hemisphere sensorimotor network structural connectivity in children with perinatal stroke is different from controls, we used diffusion imaging and graph theory to explore structural topology between these populations. Children underwent diffusion and anatomical 3T MRI. Whole-brain tractography was constrained using a brain atlas creating an adjacency matrix containing connectivity values. Graph theory metrics including betweenness centrality, clustering coefficient, and both neighbourhood and hierarchical complexity of sensorimotor nodes were compared to controls. Relationships between these connectivity metrics and validated sensorimotor assessments were explored. Eighty-five participants included 27 with venous stroke (mean age = 11.5 ± 3.7 years), 26 with arterial stroke (mean age = 12.7 ± 4.0 years), and 32 controls (mean age = 13.3 ± 3.6 years). Non-lesioned primary motor (M1), somatosensory (S1) and supplementary motor (SMA) areas demonstrated lower betweenness centrality and higher clustering coefficient in stroke groups. Clustering coefficient of M1, S1, and SMA were inversely associated with clinical motor function. Hemispheric betweenness centrality and clustering coefficient were higher in stroke groups compared to controls. Hierarchical and average neighbourhood complexity across the hemisphere were lower in stroke groups. Developmental plasticity alters the connectivity of key nodes within the sensorimotor network of the non-lesioned hemisphere following perinatal stroke and contributes to clinical disability. [ABSTRACT FROM AUTHOR]

The closed neighborhood of a vertex v 2 V (G) is, NG[v] = NG(v) [ fvg, where NG(v) is the set of all vertices which are adjacent to v. Motivated by the concept of closed neighborhood, we define a new corona called as, closed neighborhood corona. Further, we study polynomials associ- ated with adjacency matrix, Laplacian matrix and signless Laplacian matrix of the same structure. Also, we study polynomials associated with adjacency matrix, Laplacian matrix and signless Laplacian matrix of the structure of neighborhood complement corona. [ABSTRACT FROM AUTHOR]

Let G be a graph on n vertices with adjacency matrix A , and let 1 be the all-ones vector. We call G controllable if the set of vectors 1, A 1,..., An-1 1 spans the whole space Rn. We characterize the isomorphism problem of controllable graphs in terms of other combinatorial, geometric and logical problems. We also describe a polynomial time algorithm for graph isomorphism that works for almost all graphs. [ABSTRACT FROM AUTHOR]

Spectra of network related graphs have numerous applications in computer sciences, electrical networks and complex networks to explore structural characterization like stability and strength of these different real-world networks. In present article, our consideration is to compute spectrum based results of generalized prism graph which is well-known planar and polyhedral graph family belongs to the generalized Petersen graphs. Then obtained results are applied to compute some network related quantities like global mean-first passage time, average path length, number of spanning trees, graph energies and spectral radius. [ABSTRACT FROM AUTHOR]

As part of studies that employ health electronic records databases, this paper advocates the employment of graph theory for investigating drug-switching behaviors. Unlike the shared approach in this field (comparing groups that have switched with control groups), network theory can provide information about actual switching behavior patterns. After a brief and simple introduction to fundamental concepts of network theory, here we present (i) a Python script to obtain an adjacency matrix from a records database and (ii) an illustrative example of the application of network theory basic concepts to investigate drug-switching behaviors. Further potentialities of network theory (weighted matrices and the use of clustering algorithms), along with the generalization of these methods to other kinds of switching behaviors beyond drug switching, are discussed. [ABSTRACT FROM AUTHOR]

Abstract In skeleton‐based action recognition, the graph convolutional network (GCN) has achieved great success. Modelling skeleton data in a suitable spatial‐temporal way and designing the adjacency matrix are crucial aspects for GCN‐based methods to capture joint relationships. In this study, we propose the spatial‐temporal slowfast graph convolutional network (STSF‐GCN) and design the adjacency matrices for the skeleton data graphs in STSF‐GCN. STSF‐GCN contains two pathways: (1) the fast pathway is in a high frame rate, and joints of adjacent frames are unified to build ‘small’ spatial‐temporal graphs. A new spatial‐temporal adjacency matrix is proposed for these ‘small’ spatial‐temporal graphs. Ablation studies verify the effectiveness of the proposed adjacency matrix. (2) The slow pathway is in a low frame rate, and joints from all frames are unified to build one ‘big’ spatial‐temporal graph. The adjacency matrix for the ‘big’ spatial‐temporal graph is obtained by computing self‐attention coefficients of each joint. Finally, outputs from two pathways are fused to predict the action category. STSF‐GCN can efficiently capture both long‐range and short‐range spatial‐temporal joint relationships. On three datasets for skeleton‐based action recognition, STSF‐GCN can achieve state‐of‐the‐art performance with much less computational cost.

Statistical characterizations of complex network structures can be obtained from both the Ihara Zeta function (in terms of prime cycle frequencies) and the partition function from statistical mechanics. However, these two representations are usually regarded as separate tools for network analysis, without exploiting the potential synergies between them. In this paper, we establish a link between the Ihara Zeta function from algebraic graph theory and the partition function from statistical mechanics, and exploit this relationship to obtain a deeper structural characterisation of network structure. Specifically, the relationship allows us to explore the connection between the microscopic structure and the macroscopic characterisation of a network. We derive thermodynamic quantities describing the network, such as entropy, and show how these are related to the frequencies of prime cycles of various lengths. In particular, the n-th order partial derivative of the Ihara Zeta function can be used to compute the number of prime cycles in a network, which in turn is related to the partition function of Bose–Einstein statistics. The corresponding derived entropy allows us to explore a phase transition in the network structure with critical points at high and low-temperature limits. Numerical experiments and empirical data are presented to evaluate the qualitative and quantitative performance of the resulting structural network characterisations. [ABSTRACT FROM AUTHOR]

Group decision-making (GDM) is crucial in various components of graph theory, management science, and operations research. In particular, in an intuitionistic fuzzy group decision-making problem, the experts communicate their preferences using intuitionistic fuzzy preference relations (IFPRs). This approach is a way that decision-makers rank or select the most desirable alternatives by gathering criteria-based information to estimate the best alternatives using a wider range of knowledge and experience. This article proposes a new statistical measure in a fuzzy environment when the data is ambiguous or unreliable to solve a decision-making problem. This study uses the variation coefficient measure combined with intuitionistic fuzzy graphs (IFG) and Laplacian energy (LE) to solve a GDM problem that utilizes intuitionistic fuzzy preference relations (IFPRs) to select a reliable alliance partner. Initially, the Laplacian energy determines the weight of individual standards, and the obtained weight average further estimates the overall criterion weight vector. We establish the authority criteria weights using the variation coefficient measure and then ultimately rank the alternatives for each criterion using the same measure. We examine four distinct companies Alpha, Beta, Delta, and Zeta to conduct a realistic GDM to choose which alliance partner would be ideal. We successfully implemented the suggested technique, determining that Alpha satisfies company standards and is ranked first among other companies. Moreover, this technique is useful for all kinds of Intuitionistic fuzzy group decision-making problems to select optimal ones. [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 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]

Brain networks are hypothesized to undergo significant changes over development, particularly during infancy. Thus, the aim of this study is to evaluate brain maturation in the first year of life in terms of electrophysiological (EEG) functional connectivity (FC). Whole-brain FC metrics (i.e., magnitude-squared coherence, phase lag index, and parameters derived from graph theory) were extracted, for multiple frequency bands, from baseline EEG data recorded from 146 typically developing infants at 6 (T6) and 12 (T12) months of age. Generalized linear mixed models were used to test for significant differences in the computed metrics considering time point and sex as fixed effects. Correlational analyses were performed to ascertain the potential relationship between FC and subjects' cognitive and language level, assessed with the Bayley-III scale at 24 (T24) months of age. The results obtained highlighted an increased FC, for all the analyzed frequency bands, at T12 with respect to T6. Correlational analyses yielded evidence of the relationship between FC metrics at T12 and cognition. Despite some limitations, our study represents one of the first attempts to evaluate brain network evolution during the first year of life while accounting for correspondence between functional maturation and cognitive improvement. [ABSTRACT FROM AUTHOR]

In multi‐agent systems (MASs), not only do agents engage in cooperative relationships, but they also compete with one another. The topic of collaborative group in MASs has garnered significant attention. Due to the possibility of explicit interaction among agents, the complete sharing of known information among nodes presents a risk of privacy leakage. Therefore, this article explores the problem of asymmetric bipartite consensus control for MASs with strong‐privacy‐preserving. A novel state decomposition privacy‐preserving algorithm is proposed to achieve asymmetric bipartite consensus of MASs. This algorithm can ensure that the initial and final states of each agent will not be compromised, via a preservation coefficient designed in proposed algorithm. By employing algebraic graph theory and automatic control theory, a thorough analysis of the algorithmic performance is conducted. It is demonstrated that our algorithm can preserve the privacy of all agents in MASs while achieving asymmetric bipartite consensus, i.e. the consensus values are different in sign and modulus. Finally, the effectiveness of the main results is validated through simulation examples. [ABSTRACT FROM AUTHOR]

Introduction: In goal-directed tasks, visual prompts before the appearance of goals can make people ready in advance, which helps them to complete the movement better, and the presentation type of the visual prompt is very important. In previous studies, it has not been clear how different types of visual prompts guide attention in goal-directed tasks. Methods: According to the characteristics of goal-directed tasks, our research designed three different prompts: the cue prompt (featuring static arrow), the preparation prompt (involving dynamic countdown), and the combination prompt of cue and preparation information (simultaneously incorporating arrow and countdown). We used event-related potential components (CNV and P300) and graph theory indicators (clustering coefficient and characteristic path length) under the brain function connection to analyze the attention state of the brain. Results: The results showed that the combination prompts better guided the participants' sustained attention during the prompt stage, making them well prepared for the movement. Thus, after the target appeared, the participants had better executive control and achieved a faster response to the target. However, under the combination prompt, the participants consumed more attention resources during the prompt stage. Discussion: We believe that for the participants with impaired cognitive function, cue prompts or preparation prompts can be considered, which also play a role in guiding the participants' attention and helping them make motor preparations when less attention resources are consumed. This study provides a neurophysiological and behavioral foundation for the design of visual prompts in goal-directed tasks. [ABSTRACT FROM AUTHOR]

Objective: Patients with temporal lobe epilepsy (TLE) often exhibit neurocognitive disorders; however, we still know very little about the pathogenesis of cognitive impairment in patients with TLE. Therefore, our aim is to detect changes in the structural connectivity networks (SCN) of patients with TLE. Methods: Thirty-five patients with TLE were compared with 47 normal controls (NC) matched according to age, gender, handedness, and education level. All subjects underwent thin-slice T1WI scanning of the brain using a 3.0 T MRI. Then, a large-scale structural covariance network was constructed based on the gray matter volume extracted from the structural MRI. Graph theory was then used to determine the topological changes in the structural covariance network of TLE patients. Results: Although small-world networks were retained, the structural covariance network of TLE patients exhibited topological irregularities in regular architecture as evidenced by an increase in the small world properties (p < 0.001), normalized clustering coefficient (p < 0.001), and a decrease in the transfer coefficient (p < 0.001) compared with the NC group. Locally, TLE patients showed a decrease in nodal betweenness and degree in the left lingual gyrus, right middle occipital gyrus and right thalamus compared with the NC group (p < 0.05, uncorrected). The degree of structural networks in both TLE (Temporal Lobe Epilepsy) and control groups was distributed exponentially in truncated power law. In addition, the stability of random faults in the structural covariance network of TLE patients was stronger (p = 0.01), but its fault tolerance was lower (p = 0.03). Conclusion: The objective of this study is to investigate the potential neurobiological mechanisms associated with temporal lobe epilepsy through graph theoretical analysis, and to examine the topological characteristics and robustness of gray matter structural networks at the network level. [ABSTRACT FROM AUTHOR]

Against the background of the construction of new power systems, power generation, transmission, distribution, and dispatching services are open to the outside world for interaction, and the accessibility of attack paths has been significantly enhanced. We are facing cyber-physical cross-domain attacks with the characteristics of strong targeting, high concealment, and cross-space threats. This paper proposes a quantitative analysis method for the influence of power cyber-physical cross-domain attack paths based on graph knowledge. First, a layered attack graph was constructed based on the cross-space and strong coupling characteristics of the power cyber-physical system business and the vertical architecture of network security protection focusing on border protection. The attack graph included cyber-physical cross-domain attacks, control master stations, measurement and control equipment failures, transient stable node disturbances, and other vertices, and achieved a comprehensive depiction of the attack path. Second, the out-degree, in-degree, vertex betweenness, etc., of each vertex in the attack graph were comprehensively considered to calculate the vertex vulnerability, and by defining the cyber-physical coupling degree and edge weights, the risk of each attack path was analyzed in detail. Finally, the IEEE RTS79 and RTS96 node systems were selected, and the impact of risk conduction on the cascading failures of the physical space system under typical attack paths was analyzed using examples, verifying the effectiveness of the proposed method. [ABSTRACT FROM AUTHOR]

This paper investigates the long-time behavior of fractional-order complex memristive neural networks in order to analyze the synchronization of both anatomical and functional brain networks, for predicting therapy response, and ensuring safe diagnostic and treatments of neurological disorder (such as epilepsy, Alzheimer's disease, or Parkinson's disease). A new mathematical brain connectivity model, taking into account the memory characteristics of neurons and their past history, the heterogeneity of brain tissue, and the local anisotropy of cell diffusion, is proposed. This developed model, which depends on topology, interactions, and local dynamics, is a set of coupled nonlinear Caputo fractional reaction–diffusion equations, in the shape of a fractional-order ODE coupled with a set of time fractional-order PDEs, interacting via an asymmetric complex network. In order to introduce into the model the connection structure between neurons (or brain regions), the graph theory, in which the discrete Laplacian matrix of the communication graph plays a fundamental role, is considered. The existence of an absorbing set in state spaces for system is discussed, and then the dissipative dynamics result, with absorbing sets, is proved. Finally, some Mittag–Leffler synchronization results are established for this complex memristive neural network under certain threshold values of coupling forces, memristive weight coefficients, and diffusion coefficients. [ABSTRACT FROM AUTHOR]