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Designing homogeneous networks is considered one typical strategy for solving the problem of strength and toughness conflict of polymer network materials. Experimentalists have proposed the hypothesis of obtaining a structurally homogeneous hydrogel by crosslinking tetra-armed polymers, whose homogeneity was claimed to be verified by scattering characterization and other methods. Nevertheless, it is highly desirable to further evaluate this issue from other perspectives. In this study, a coarse-grained molecular dynamics simulation coupled with a stochastic reaction model is applied to reveal the topological structure of a polymer network synthesized by tetra-armed monomers as precursors. Two different scenarios, distinguished by whether internal cross-linking is allowed, are considered. We introduce the Dijkstra algorithm from graph theory to precisely characterize the network structure. The microscopic features of the network structure, e.g., loop size, dispersity, and size distribution, are obtained via the Dijkstra algorithm. By comparing the two reaction scenarios, Scenario II exhibits an overall more idealized structure. Our results demonstrate the feasibility of the Dijkstra algorithm for precisely characterizing the polymer network structure. We expect this work will provide a new insight for the evaluation and description of gel networks and further help to reveal the dynamic process of network formation. [ABSTRACT FROM AUTHOR]

Biomolecular condensates formed by multicomponent phase separation play crucial roles in diverse cellular processes. Accurate assessment of individual-molecule contributions to condensate formation and precise characterization of their spatial organization within condensates are crucial for understanding the underlying mechanism of phase separation. Using molecular dynamics simulations and graph theoretical analysis, we demonstrated quantitatively the significant roles of cation-π and π–π interactions mediated by aromatic residues and arginine in the formation of condensates in polypeptide systems. Our findings reveal temperature and chain length-dependent alterations in condensate network parameters, such as the number of condensate network layers, and changes in aggregation and connectivity. Notably, we observe a transition between assortativity and disassortativity in the condensate network. Moreover, polypeptides W, Y, F, and R consistently promote condensate formation, while the contributions of other charged and two polar polypeptides (Q and N) to condensate formation depend on temperature and chain length. Furthermore, polyadenosine and polyguanosine can establish stable connections with aromatic and R polypeptides, resulting in the reduced involvement of K, E, D, Q, and N in phase separation. Overall, this study provides a distinctive, precise, and quantitative approach to characterize the multicomponent phase separation. [ABSTRACT FROM AUTHOR]

Modeling the dynamics of glassy systems has been challenging in physics for several decades. Recent studies have shown the efficacy of Graph Neural Networks (GNNs) in capturing particle dynamics from the graph structure of glassy systems. However, current GNN methods do not take the dynamic patterns established by neighboring particles explicitly into account. In contrast to these approaches, this paper introduces a novel dynamical parameter termed "smoothness" based on the theory of graph signal processing, which explores the dynamic patterns from a graph perspective. Present graph-based approaches encode structural features without considering smoothness constraints, leading to a weakened correlation between structure and dynamics, particularly on short timescales. To address this limitation, we propose a Geometry-enhanced Graph Neural Network (Geo-GNN) to learn the smoothness of dynamics. Results demonstrate that our method outperforms state-of-the-art baselines in predicting glassy dynamics. Ablation studies validate the effectiveness of each proposed component in capturing smoothness within dynamics. These findings contribute to a deeper understanding of the interplay between glassy dynamics and static structure. [ABSTRACT FROM AUTHOR]

In this paper, we present a new module to predict the potential surface reconstruction configurations of given surface structures in the framework of our machine learning and graph theory assisted universal structure searcher. In addition to random structures generated with specific lattice symmetry, we made full use of bulk materials to obtain a better distribution of population energy, namely, randomly appending atoms to a surface cleaved from bulk structures or moving/removing some of the atoms on the surface, which is inspired by natural surface reconstruction processes. In addition, we borrowed ideas from cluster predictions to spread structures better between different compositions, considering that surface models of different atom numbers usually have some building blocks in common. To validate this newly developed module, we tested it with studies on the surface reconstructions of Si (100), Si (111), and 4H–SiC (1 1 ̄ 02) − c (2 × 2) , respectively. We successfully gave the known ground states, as well as a new SiC surface model, in an extremely Si-rich environment. [ABSTRACT FROM AUTHOR]

The normalized distance Laplacian of a graph G is defined as D L (G) = T (G) − 1 / 2 (T (G) − D (G)) T (G) − 1 / 2 where D (G) is the matrix with pairwise distances between vertices and T (G) is the diagonal transmission matrix. In this project, we study the minimum and maximum spectral radii associated with this matrix, and the structures of the graphs that achieve these values. In particular, we prove a conjecture of Reinhart that the complete graph is the unique graph with minimum spectral radius, and we give several partial results towards a second conjecture of Reinhart regarding which graph has the maximum spectral radius. [ABSTRACT FROM AUTHOR]

PROFESSIONAL MUSICIANS HAVE BEEN TEACHING/ learning/interpreting Western classical tonal music for longer than atonal music. This may be reflected in their brain plasticity and playing efficiency. To test this idea, EEG connectivity networks (EEG-CNs) of expert cellists at rest and during real and imagined musical interpretation of tonal and atonal excerpts were analyzed. Graphs and connectomes were constructed as models of EEG-CNs, using functional connectivity measurements of EEG phase synchronization in different frequency bands. Tonal and atonal interpretation resulted in a global desynchronization/dysconnectivity versus resting--irrespective of frequency bands--particularly during imagined-interpretation. During the latter, the normalized local information-transfer efficiency (NLE) of graph-EEG-CN's small-world structure at rest increased significantly during both tonal and atonal interpretation, and more significantly during atonalinterpretation. Regional results from the graphs/connectomes supported previous findings, but only certain EEG frequency bands. During imagined-interpretation, the number of disconnected regions and subnetworks, as well as regions with higher NLE, were greater in atonal-interpretation than in tonal-interpretation for delta/theta/gamma-EEG-CNs. The opposite was true during real-interpretation, specifically limited to alpha-EEG-CN. Our EEG-CN experimental paradigm revealed perceptual differences in musicians' brains during tonal and atonal interpretations, particularly during imagined-interpretation, potentially due to differences in cognitive roots and brain plasticity for tonal and atonal music, which may affect the musicians' interpretation. [ABSTRACT FROM AUTHOR]

• Multiple sclerosis can lead to various functional alterations of sensorimotor brain networks. • The value of resting-state EEG or dynamic EEG during a motor tastk to assess motor impaiment in multiple sclerosis was reviewed. • EEG biomarkers could be used to develop future neuromodulation therapy of disease-related motor disorders in multiple sclerosis. Electroencephalography (EEG) can highlight significant changes in spontaneous electrical activity of the brain produced by altered brain network connectivity linked to inflammatory demyelinating lesions and neuronal loss occurring in multiple sclerosis (MS). In this review, we describe the main EEG findings reported in the literature to characterize motor network alteration in term of local activity or functional connectivity changes in patients with MS (pwMS). A comprehensive literature search was conducted to include articles with quantitative analyses of resting-state EEG recordings (spectrograms or advanced methods for assessing spatial and temporal dynamics, such as coherence, theory of graphs, recurrent quantification, microstates) or dynamic EEG recordings during a motor task, with or without connectivity analyses. In this systematic review, we identified 26 original articles using EEG in the evaluation of MS-related motor disorders. Various resting or dynamic EEG parameters could serve as diagnostic biomarkers of motor control impairment to differentiate pwMS from healthy subjects or be related to a specific clinical condition (fatigue) or neuroradiological aspects (lesion load). We highlight some key EEG patterns in pwMS at rest and during movement, both suggesting an alteration or disruption of brain connectivity, more specifically involving sensorimotor networks. Some of these EEG biomarkers of motor disturbance could be used to design future therapeutic strategies in MS based on neuromodulation approaches, or to predict the effects of motor training and rehabilitation in pwMS. [ABSTRACT FROM AUTHOR]

The ranking of nodes in a network according to their centrality or ''importance'' is a classic problem that has attracted the interest of different scientific communities in the last decades. The COVID-19 pandemic has recently rejuvenated the interest in this problem, as the ranking may be used to decide who should be tested, or vaccinated, first, in a population of asymptomatic individuals. In this paper, we review classic methods for node ranking and compare their performance in a benchmark network that considers the community-based structure of society. The outcome of the ranking procedure is then used to decide which individuals should be tested, and possibly quarantined, first. Finally, we also review the extension of these ranking methods to weighted graphs and explore the importance of weights in a contact network by providing a toy model and comparing node rankings for this case in the context of disease spread. [ABSTRACT FROM AUTHOR]

With the emerging era of E‐commerce and online shopping, people are also in a habit to receive default product recommendations on the web pages that they access. Google is already providing such suggestions. Till now recommendations were made only based on previous sentiments or feedback or ratings, but this research has improved the product recommendation method by including one more parameter for the same. This article represents two parameters for making predictions of product allocation to a new customer. These parameters are ratings given by the existing users for that particular product and the region to which the new customer belongs. Following these parameters, a prediction model and an algorithm, Improved_Collab_Similarity, have been implemented. The dataset has been developed where India as a country along with all its States has been considered for products which are popular for their creation based on regional and ancient skills of the people belonging to that area. Results for the mentioned prediction model have been discussed in this article where generally precision increases with the increase in a number of products but at some points, it does not increase when a smaller number of that product was purchased by the customers. [ABSTRACT FROM AUTHOR]

The concept of rainbow connection was introduced by Chartrand et al. in 2008. The k -rainbow index, which is a generalization of rainbow connection number was also introduced by Chartrand et al. in 2010. These two new graph-parameters are defined for all edge-colored graphs. Krivelevich and Yuster in 2010 naturally introduced the rainbow vertex-connection number which is defined on vertex-colored graphs. As a natural counterpart of the k -rainbow index and a generalization of rainbow vertex-connection number, Mao introduced the concept of k -vertex-rainbow index in 2016. Nowadays they have become new and active subjects in graph theory. In this survey we attempt to bring together most of the basic and new results that dealt with the k -rainbow index and k -vertex-rainbow index. [ABSTRACT FROM AUTHOR]

Indium phosphide (InP) is a binary semiconductor composed of indium and phosphorus. It has a zinc blende crystal structure, which is a type of cubic lattice structure. This lattice is composed of indium and phosphorus atoms arranged in a lattice of cube-shaped cells, with each cell containing four indium atoms and four phosphorus atoms. This lattice structure is the same for all materials with a zinc blende crystal structure and is the most common type of lattice structure in semiconductors. Indium phosphide (InP) has several chemical applications. It is commonly used as a dopant in the production of semiconductors, where it helps control the electrical properties of the material. InP is also utilized in the synthesis various indium-containing compounds, which can have applications in catalysts and chemical reactions. Additionally, InP nanoparticles have been investigated for their potential use in biomedical imaging and drug delivery systems. The topological characterization of 3D molecular structures can be performed via graph theory. In graph theory, the connections between atoms are represented as edges and the atoms themselves are represented as nodes. Furthermore, graph theory can be used to calculate the topological descriptors of the molecule such as the degree-based and reverse degree-based irregularity toplogical indices. These descriptors can be used to compare the topology of different molecules. This paper deals with the modeling and topological characterization of indium phosphide (InP) via degree-based and reverse irregularity indices. The 3D crystal structure of the InP is topologically modeled via partition of the edges, and derived closed form expressions for its irregularity indices. Our obtained results will be surely be helpful in investigating the QSPR/QSAR analysis as well as understanding the deep irregular behavior of the indium phosphide (InP) . [ABSTRACT FROM AUTHOR]

Under the envisioned smart grid paradigm, there is an increasing demand for a fast, accurate, and efficient power flow solution for distribution system operation and control. Various solution techniques have been proposed, each with its own unique formulation, solution methodology, advantages, and drawbacks. Motivated by challenges associated with the integration of renewable distributed energy resources and electric vehicles into distribution systems and further by the speed and convergence limitations of existing tools, this paper presents a novel graph‐based power flow solution for smart grid's real‐time operation and control, named Flow‐AugmentationPF algorithm. The proposed method formulates a power flow problem as a network‐flow problem and solves it by using a maximum‐flow algorithm, inspired by the push‐relabel max‐flow technique. The performance of the proposed algorithm is tested and validated using several benchmark networks of different sizes, topologies, and parameters and compared against the most commonly used solution techniques and commercial software packages, namely PSS/E and PSCAD. The proposed formulation is simple, accurate, fast, yet computationally efficient, as it is based on matrix‐vector multiplication, and is also scalable, considering the formulation works as a graph‐based method, which, inherently, allows for parallel computation for added computational speed. [ABSTRACT FROM AUTHOR]

Collaborative crowdsensing is a team collaboration model that harnesses the intelligence of a large network of participants, primarily applied in areas such as intelligent computing, federated learning, and blockchain. Unlike traditional crowdsensing, user recruitment in collaborative crowdsensing not only considers the individual capabilities of users but also emphasizes their collaborative abilities. In this context, this paper takes a unique approach by modeling user interactions as a graph, transforming the recruitment challenge into a graph theory problem. The methodology employs an enhanced Prim algorithm to identify optimal team members by finding the maximum spanning tree within the user interaction graph. After the recruitment, the collaborative crowdsensing explored in this paper presents a challenge of unfair incentives due to users engaging in free-riding behavior. To address these challenges, the paper introduces the MR-SVIM mechanism. Initially, the process begins with a Gaussian mixture model predicting the quality of users' tasks, combined with historical reputation values to calculate their direct reputation. Subsequently, to assess users' significance within the team, aggregation functions and the improved PageRank algorithm are employed for local and global influence evaluation, respectively. Indirect reputation is determined based on users' importance and similarity with interacting peers. Considering the comprehensive reputation value derived from the combined assessment of direct and indirect reputations, and integrating the collaborative capabilities among users, we have formulated a feature function for contribution. This function is applied within an enhanced Shapley value method to assess the relative contributions of each user, achieving a more equitable distribution of earnings. Finally, experiments conducted on real datasets validate the fairness of this mechanism. [ABSTRACT FROM AUTHOR]

This paper is concerned with the synchronization of stochastic uncertain complex dynamic networks with time‐varying delays. In contrast to existing synchronization network models, the current study considers both internal time‐varying delays and coupling time‐varying delays. By analyzing the two factors (i.e. time‐varying delays and uncertainties) on the synchronization performance of stochastic uncertain complex dynamic networks, a new inverse optimal adaptive control is proposed. The analysis tool developed in this paper is based on Lyapunov technique, algebraic graph theory and inverse optimality, which does not need to solve Hamilton‐Jacobi‐Bellman equation. Some sufficient conditions are established for the synchronization of stochastic uncertain complex dynamic networks with time‐varying delays. To demonstrate the effectiveness of the present approach, simulation results are illustrated. In this paper, we consider the synchronization of stochastic uncertain complex dynamic networks with time‐varying delays. Different from the existing synchronization of the networks model, we take the internal time‐varying delays and the coupling time‐varying delays into account. By analyzing the two factors (i.e. time‐varying delays and uncertainties) on the synchronization performance of stochastic uncertain complex dynamic networks, a novel inverse optimal adaptive control is proposed. The analysis tool developed in this paper is based on Lyapunov technique, algebraic graph theory and inverse optimality, which does not need to solve Hamilton‐Jacobi‐Bellman equation. Some sufficient conditions are established for the synchronization of stochastic uncertain complex dynamic networks with time‐varying delays. [ABSTRACT FROM AUTHOR]

Given a connected graph G = (V , E) , with n ≔ | V | vertices and m ≔ | E | edges, a cut can be represented as a bipartition { S , S ¯ } of the vertices or as the set of those edges in E having one endpoint in S and the other in S ¯ , denoted by δ G (S , S ¯). A cut is minimal, or also called bond , if and only if the two induced subgraphs obtained by removing the edges in the cut are both connected. When the bond separates two given vertices s and t , we talk about s , t - bond. In this work, we consider the problems of listing all the bonds and listing all the s , t -bonds in a graph. These fundamental problems find application in many research areas, such as, beyond graph theory, network reliability, bioinformatics, and chemistry. The state-of-the-art algorithm exploits binary partition to output each s , t -bond in O (m) per bond, being thus classified as an O (m) -delay algorithm. Here we present two new algorithms to address these problems. The first one implements a slightly different branching strategy than the state of the art, though achieving the same complexity. Anyway, we can improve it by relying on dynamic data structures and amortized analysis, obtaining an algorithm that outputs a new bond in O ˜ (n). By assuming only the two bond-shores are output for every bond, this is the first output-linear algorithm listing bonds. In case we commit to explicitly providing the entire edge-set of every bond, the delay becomes O ˜ (n) + | δ G (S , S ¯) |. • We study the problem of listing all minimal cuts (or bonds) in a graph. • We exploit a different branching strategy than Tsukiyama et al. (1980). • We provide a first algorithm having the same complexity as Tsukiyama et al. (1980). • We rely on dynamic data structures and amortized analysis to improve our approach. • We propose the first O ˜ (n) -delay algorithm to list all bonds in a graph. [ABSTRACT FROM AUTHOR]

Numerous graph-theoretic methods can be used to investigate the efficiency and reliability of networks. The reliability of a network is determined based on its connectivity. At the very least, the system must continue to function in the event of part of its component (node or link) failures. An ideal system remains reliable even when errors occur. Diameter is a measure of network efficiency used to study the effects of link failures in networks. In this paper, we look into the eliminating edges affect to the hypercube graph’s diameter (call it by d) and calculate the maximum diameter that is denoted by f(t,d) after deleting t edges from hypercube graphs Qn for some n. [ABSTRACT FROM AUTHOR]

Approximate concept and graph theory are developed and applied to solve the integrated optimization problem of engineering structure containing both beam and plate discrete variables. Firstly, both structural responses and explicit requirements for process manufacturing are considered as constraints, which makes an optimization system based on the two-level multi-point approximation method more engineering oriented. Secondly, a practical evaluation method of topology rationality is proposed based on graph theory in order to avoid invalid structural analysis and improve optimization efficiency. The finite element model corresponding to the binary point-line relation in graph theory and judgements of force nodes, lump masses, structural integrity, etc. are unified. An engineering example shows that the two-level multi-point approximation method is still efficient in solving topology optimization problems with participating plates and beams, and the proposed topological rationality evaluation method can effectively eliminate about 15% of unreasonable configurations. [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]

This paper introduces a virtual consensus‐based wide area differential protection method through cooperative control concepts and graph theory. Using multi‐agent systems eliminates the need for a direct telecommunication connection between the protective relays to implement the proposed differential protection. In addition, applying telecommunication subgraphs facilitates the establishment of numerous differential protection areas. Collaboration between protected areas is facilitated by common agents, establishing a wide‐area cooperative protection network. The capability of the network operator to define various protection areas and the collaboration between these areas ensure the versatility of the proposed method for various protection purposes. The present paper primarily represents the application of the proposed protection system as a wide‐area supervisor protection for distance relays. Performance evaluations on an IEEE 39‐bus test system illustrate the method's effectiveness in various scenarios. The results show enhanced power system stability and resilience, particularly when traditional methods struggle to detect power swings with high impedance variation rates. [ABSTRACT FROM AUTHOR]

Summary: In this article, the cluster consensus problems of two classes of nonlinear multi‐agent systems (NMASs) are studied under the dynamic event‐triggered control (ETC) strategy, where cooperative‐competitive interactions among agents and high‐order dynamics are considered. First, two dynamic ETC protocols are developed to realize the cluster synchronization of NMASs. Different from the static ETC, a dynamic parameter regulated by an adaptive law is used in place of a constant which can improve the trigger time intervals and reduce the energy consumption. Subsequently, an adaptive control mechanism based on the proposed ETC strategy is designed to adjust the controller parameter for agents. Some sufficient conditions for achieving the cluster consensus of NMASs are derived by using the Lyapunov stability theory and algebraic graph theory. Moreover, it is proved in theory that Zeno phenomenon can be excluded with these control schemes. Finally, two simulation examples validate the effectiveness of the theoretical results. [ABSTRACT FROM AUTHOR]

Multiple kernel learning (MKL) is an efficient way to improve hyperspectral image classification with few training samples by integrating spectral and spatial features. Nonetheless, presenting a MKL algorithm to capture significant information and eliminate redundancy is indispensable. In this paper, a novel graph-based MKL method (GMKL) is introduced to enhance the performance of the MKL algorithm and achieve high discriminative features, resulting in maximum separability. To do so, an informative low-rank graph is determined via a new algorithm called block diagonal representation that takes into account both local and global data structures. Then, in lieu of a time-consuming search for finding the optimal combination of the basic kernels, unlike the prior works, an optimal projective direction is acquired using graph theory. Various basic kernels generated from superpixels and attribute profiles are projected to produce a discriminative combination of kernels. Eventually, the optimum kernel is given to the support vector machine classifier. Experiments have been conducted on two popular hyperspectral datasets (Indian Pines and Pavia University). For the datasets, the proposed algorithm gained 90.43% and 99.28% overall accuracy, respectively, using only 1% training samples, which is much better than the state-of-the-art MKL algorithm and deep learning-based approaches with lower computational time. [ABSTRACT FROM AUTHOR]

Isolated rapid-eye-movement behavior disorder (iRBD) often precedes the development of alpha-synucleinopathies such as Parkinson's disease (PD). Magnetic resonance imaging (MRI) studies have revealed structural brain alterations in iRBD partially resembling those observed in PD. However, relatively little is known about whole-brain functional brain alterations in iRBD. Here, we characterize the functional brain connectome of iRBD compared with PD patients and healthy controls (HC) using resting-state functional MRI (rs-fMRI). Eighteen iRBD subjects (67.3 ± 6.6 years), 18 subjects with PD (65.4 ± 5.8 years), and 39 age- and sex-matched HC (64.4 ± 9.2 years) underwent rs-fMRI at 3 T. We applied a graph theoretical approach to analyze the brain functional connectome at the global and regional levels. Data were analyzed using both frequentist and Bayesian statistics. Global connectivity was largely preserved in iRBD and PD individuals. In contrast, both disease groups displayed altered local connectivity mainly in the motor network, temporal cortical regions including the limbic system, and the visual system. There were some group specific alterations, and connectivity changes were pronounced in PD individuals. Overall, however, there was a good agreement of the connectome changes observed in both disease groups. This study provides evidence for widespread functional brain connectivity alterations in iRBD, including motor circuitry, despite normal motor function. Connectome alterations showed substantial resemblance with those observed in PD, underlining a close pathophysiological relationship of iRBD and PD. • Isolated REM-Sleep-behavior-disorder (iRBD) is a prodrome of Parkinson's disease (PD). • IRBD is characterized by wide spread functional brain alterations. • These connectivity changes show remarkable resemblance with those found in PD. • Our work suggests a close pathophysiological relationship of iRBD with PD. [ABSTRACT FROM AUTHOR]

The unique organizational framework for polycyclic aromatic hydrocarbons developed by us has led to the discovery of number patterns for their number of isomers. Constant-isomer series are generated by repetitive circumscribing of a given set of isomers. Benzenoid, fluoranthenoid, indacenoid, and primitive coronoid constant isomer series possess twin isomer numbers with members having corresponding topologies. The concepts of strictly pericondensed, strain-free, Clar's aromatic sextet, and symmetry are interconnected in the topological correspondence between strictly pericondensed and total resonant sextet (TRS) benzenoid hydrocarbons. In a plot of TRS isomer numbers on the Formula Periodic Table for Total Resonant Sextet Benzenoids [Table PAH6(TRS)] which is a subset of Table PAH6 for ordinary benzenoids, structural correlations in isomer numbers, symmetry distributions, and empty rings between various strain-free TRS benzenoids are presented. These chemical graph theoretical results belong to the branch of mathematics called number theory. [ABSTRACT FROM AUTHOR]

Given a graph G $G$, we let s+(G) ${s}^{+}(G)$ denote the sum of the squares of the positive eigenvalues of the adjacency matrix of G $G$, and we similarly define s−(G) ${s}^{-}(G)$. We prove that χf(G)≥1+maxs+(G)s−(G),s−(G)s+(G) ${\chi }_{f}(G)\ge 1+\max \left\{\frac{{s}^{+}(G)}{{s}^{-}(G)},\frac{{s}^{-}(G)}{{s}^{+}(G)}\right\}$ and thus strengthen a result of Ando and Lin, who showed the same lower bound for the chromatic number χ(G) $\chi (G)$. We in fact show a stronger result wherein we give a bound using the eigenvalues of G $G$ and H $H$ whenever G $G$ has a homomorphism to an edge‐transitive graph H $H$. Our proof utilizes ideas motivated by association schemes. [ABSTRACT FROM AUTHOR]

The graph removal lemma is a fundamental result in extremal graph theory which says that for every fixed graph H and ε > 0 , if an n -vertex graph G contains ε n 2 edge-disjoint copies of H then G contains δ n v (H) copies of H for some δ = δ (ε , H) > 0. The current proofs of the removal lemma give only very weak bounds on δ (ε , H) , and it is also known that δ (ε , H) is not polynomial in ε unless H is bipartite. Recently, Fox and Wigderson initiated the study of minimum degree conditions guaranteeing that δ (ε , H) depends polynomially or linearly on ε. In this paper we answer several questions of Fox and Wigderson on this topic. [ABSTRACT FROM AUTHOR]

A graph H is common if the number of monochromatic copies of H in a 2-edge-colouring of the complete graph K n is asymptotically minimised by the random colouring, or equivalently, t H (W) + t H (1 − W) ≥ 2 1 − e (H) holds for every graphon W : [ 0 , 1 ] 2 → [ 0 , 1 ] , where t H (.) denotes the homomorphism density of the graph H. Paths and cycles being common is one of the earliest cornerstones in extremal graph theory, due to Mulholland and Smith (1959), Goodman (1959), and Sidorenko (1989). We prove a graph homomorphism inequality that extends the commonality of paths and cycles. Namely, t H (W) + t H (1 − W) ≥ t K 2 (W) e (H) + t K 2 (1 − W) e (H) whenever H is a path or a cycle and W : [ 0 , 1 ] 2 → R is a bounded symmetric measurable function. This answers a question of Sidorenko from 1989, who proved a slightly weaker result for even-length paths to prove the commonality of odd cycles. Furthermore, it also settles a recent conjecture of Behague, Morrison, and Noel in a strong form, who asked if the inequality holds for graphons W and odd cycles H. Our proof uses Schur convexity of complete homogeneous symmetric functions, which may be of independent interest. [ABSTRACT FROM AUTHOR]

• We introduce a graph theory model and utilize dispatchable STN to address the mission planning problem for deep space detectors. • The proposed algorithm efficiently decreases the network scale and streamline the computational workload. • Our proposed algorithm effectively tackles the challenge of dynamic mission scheduling to effectively handle emergencies. Deep space exploration activities can effectively promote the development and application of space technology, which holds significant scientific and strategic value. The multi-mission and long-period detector is executed in deep space, and numerous temporal constraints are generated by mission planning. Moreover, scheduled missions may face external uncertainties, hindering their smooth execution. Therefore, temporal constraint processing strategies that possess dynamic adjustment capabilities play a crucial role in ensuring the successful execution of deep space detectors. Existing methods primarily concentrate on investigating fixed mission series, with limited efficiency in planning. Specifically, these methods do not rigorously enforce temporal constraints and lack the capability to handle the disruptions caused by emergencies, which hinder the smooth execution of planned missions. In this paper, we propose a dynamic temporal constraints algorithm, aiming to efficiently plan missions subject to temporal constraints and let it possess a certain degree of dynamic adjust capability. The simulation results demonstrate the effective simplification of redundant temporal constraints in mission networks by the proposed algorithm. In comparison to the classical algorithm, the computation time exhibits a clear advantage. Furthermore, the proposed algorithm possesses the capability to perform local, small-scale dynamic adjustments in response to emergencies that may disrupt the normal execution of missions. Consequently, this research establishes a robust basis for addressing temporal constraints in deep space detectors. [ABSTRACT FROM AUTHOR]

This article proposes a new distributed finite‐time optimization algorithm for agents under directed graphs. By employing the nonsmooth technique and graph theory, a distributed discontinuous algorithm for continuous‐time agents subject to strongly convex local cost functions is first designed with a finite‐time distributed estimator, where the gradients of the local cost functions are estimated in finite time. It is shown that for a strongly connected graph and arbitrary initial conditions, the proposed algorithms can achieve consensus, and the systems can converge to the optimal point in finite time. Then, a two‐step approach is proposed to achieve finite‐time optimization of high‐order agents with disturbances under directed graphs. Finally, the validity of the proposed finite‐time optimization algorithm is verified by two numerical examples. [ABSTRACT FROM AUTHOR]

Partition problems in graphs are extremely important in applications, as shown in the Data Science and Machine Learning literature. One approach is spectral partitioning based on a Fiedler vector, i.e., an eigenvector corresponding to the second smallest eigenvalue a (G) of the Laplacian matrix L G of the graph G. This problem corresponds to the minimization of a quadratic form associated with L G , under certain constraints involving the ℓ 2 -norm. We introduce and investigate a similar problem, but using the ℓ 1 -norm to measure distances. This leads to a new parameter b (G) as the optimal value. We show that a well-known cut problem arises in this approach, namely the sparsest cut problem. We prove connectivity results and different bounds on this new parameter, relate to Fiedler theory and show explicit expressions for b (G) for trees. We also comment on an ℓ ∞ -norm version of the problem. [ABSTRACT FROM AUTHOR]

In this article we give a geometrical description of the (in general non-selfadjoint) in/out Laplacian L + / − = (d + / −) ⁎ d and adjacency matrix on digraphs with arbitrary weights, where (d + / −) ⁎ is the adjoint of the evaluation map d + / − on the terminal/initial vertex of each arc and d = d + + d − denotes the discrete gradient. We prove that the multiplicity of the zero eigenvalue of L + / − = (d + / −) ⁎ d coincides with the number of sources/sinks of the digraph. We also show that for an acyclic digraph with combinatorial weights the spectrum is contained in the set of non-zero integers. The geometrical perspective allows to interpret the set of circulations C of a weighted digraph as coclosed forms on the arcs, i.e. as the kernel of the discrete divergence d ⁎. Moreover, C is perpendicular to the set of discrete gradients of functions on the vertices. We also give formulas to compute the capacity of a cut and the value of a flow in terms of L − and d. We illustrate the results with many concrete examples. [ABSTRACT FROM AUTHOR]

Graph pebbling is a combinatorial game played on an undirected graph with an initial configuration of pebbles. A pebbling move consists of removing two pebbles from one vertex and placing one pebble on an adjacent vertex. The pebbling number of a graph is the smallest number of pebbles necessary such that, given any initial configuration of pebbles, at least one pebble can be moved to a specified root vertex. Recent lines of inquiry apply computational techniques to pebbling bound generation and improvement. Along these lines, we present a computational framework that produces a set of tree strategy weight functions that are capable of proving pebbling number upper bounds on a connected graph. Our mixed-integer linear programming approach automates the generation of large sets of such functions and provides verifiable certificates of pebbling number upper bounds. The framework is capable of producing verifiable pebbling bounds on any connected graph, regardless of its structure or pebbling properties. We apply the model to the 4th weak Bruhat to prove π (B 4) ≤ 66 and to the Lemke square graph to produce a set of certificates that verify π (L □ L) ≤ 96. [ABSTRACT FROM AUTHOR]

Topological indices (TIs) have several biological applications in the treatment of blood cancer. TIs can be used to predict the efficacy of drugs in cancer treatment by providing information about the molecular structures of the drugs and their related properties. By analyzing the degree-based TIs of different drugs, researchers can identify the most effective drugs for treating specific types of blood cancer. Additionally, TIs can also be used to predict the toxicity of drugs, which is essential for developing safe and effective treatments. Furthermore, TIs can aid in the discovery and design of new drugs by providing insights into the relationship between molecular structure and biological activity. This article focuses on the applications of topological indices (TIs) in predicting the physical and biological properties of novel drugs used in the treatment of blood cancer. The drugs glasdegib, palbociclib, and daunorubicin are analyzed using degree-based topological indices computed through edge partitioning. A quantitative structure-property relationship (QSPR) model is then developed using linear regression to predict the properties such as boiling point (BP), flash point (FP), molar volume (MV), molecular weight, and complexity. The results demonstrate the potential of topological indices as a tool for drug discovery and design in the field of cancer treatment. [ABSTRACT FROM AUTHOR]

Built upon the shoulders of graph theory, the field of complex networks has become a central tool for studying real systems across various fields of research. Represented as graphs, different systems can be studied using the same analysis methods, which allows for their comparison. Here, we challenge the widespread idea that graph theory is a universal analysis tool, uniformly applicable to any kind of network data. Instead, we show that many classical graph metrics—including degree, clustering coefficient, and geodesic distance—arise from a common hidden propagation model: the discrete cascade. From this perspective, graph metrics are no longer regarded as combinatorial measures of the graph but as spatiotemporal properties of the network dynamics unfolded at different temporal scales. Once graph theory is seen as a model-based (and not a purely data-driven) analysis tool, we can freely or intentionally replace the discrete cascade by other canonical propagation models and define new network metrics. This opens the opportunity to design—explicitly and transparently—dedicated analyses for different types of real networks by choosing a propagation model that matches their individual constraints. In this way, we take stand that network topology cannot always be abstracted independently from network dynamics but shall be jointly studied, which is key for the interpretability of the analyses. The model-based perspective here proposed serves to integrate into a common context both the classical graph analysis and the more recent network metrics defined in the literature which were, directly or indirectly, inspired by propagation phenomena on networks. [ABSTRACT FROM AUTHOR]

Based on a constructive generalization of the results of works on the topology of phase diagrams of multi-component condensed systems, the contribution of these studies to the general theory of phase diagrams is presented. General fundamental specific features characteristic of certain classes of diagrams were identified as a result of the systematic study of this topology. The topology of subsolidus isobaric-isothermal phase diagrams, as well as melting diagrams, is considered. This review can be useful for the formulation and solution of new problems in the study of phase diagrams of multi-component systems. [ABSTRACT FROM AUTHOR]

Aims: Subsyndromal depression (SSD) is common in mild cognitive impairment (MCI). However, the neural mechanisms underlying MCI with SSD (MCID) are unclear. The default mode network (DMN) is associated with cognitive processes and depressive symptoms. Therefore, we aimed to explore the topological organization of the DMN in patients with MCID. Methods: Forty‐two MCID patients, 34 MCI patients without SSD (MCIND), and 36 matched healthy controls (HCs) were enrolled. The resting‐state functional connectivity of the DMN of the participants was analyzed using a graph theoretical approach. Correlation analyses of network topological metrics, depressive symptoms, and cognitive function were conducted. Moreover, support vector machine (SVM) models were constructed based on topological metrics to distinguish MCID from MCIND. Finally, we used 10 repeats of 5‐fold cross‐validation for performance verification. Results: We found that the global efficiency and nodal efficiency of the left anterior medial prefrontal cortex (aMPFC) of the MCID group were significantly lower than the MCIND group. Moreover, small‐worldness and global efficiency were negatively correlated with depressive symptoms in MCID, and the nodal efficiency of the left lateral temporal cortex and left aMPFC was positively correlated with cognitive function in MCID. In cross‐validation, the SVM model had an accuracy of 0.83 [95% CI 0.79–0.87], a sensitivity of 0.88 [95% CI 0.86–0.90], a specificity of 0.75 [95% CI 0.72–0.78] and an area under the curve of 0.88 [95% CI 0.85–0.91]. Conclusions: The coexistence of MCI and SSD was associated with the greatest disrupted topological organization of the DMN. The network topological metrics could identify MCID and serve as biomarkers of different clinical phenotypic presentations of MCI. [ABSTRACT FROM AUTHOR]

We investigate the zero-error coding for computing problems with encoder side information. An encoder provides access to a source X and is furnished with side information g (Y) . It communicates with a decoder that possesses side information Y and aims to retrieve f (X , Y) with zero probability of error, where f and g are assumed to be deterministic functions. In previous work, we determined a condition that yields an analytic expression for the optimal rate R * (g) ; in particular, it covers the case where P X , Y is full support. In this article, we review this result and study the side information design problem, which consists of finding the best trade-offs between the quality of the encoder's side information g (Y) and R * (g) . We construct two greedy algorithms that give an achievable set of points in the side information design problem, based on partition refining and coarsening. One of them runs in polynomial time. [ABSTRACT FROM AUTHOR]

The traditional Buck-Boost equalization circuit has long transmission path and low equalization efficiency, and the existing ring equalizer cannot fundamentally solve this problem. In view of this problem, a double-layer ring equalizer for lithium battery based on Buck-Boost and switched capacitor is proposed in this study. The equalization circuit adopts modular balancing method, and adopts Buck-Boost and switching capacitor hybrid circuit to achieve equalization in the bottom module and top module. The circuit structure is analyzed from two aspects of equalization speed and efficiency by graph theory analysis. The results show that the equalization circuit has the advantages of less energy transmission paths and high energy transmission efficiency. The simulation model of the equalization circuit is built on the MATLAB/Simulink power simulation platform. The simulation results show that the equalization speed of the double-layer ring equalizer based on Buck-Boost and switched capacitor is 46% higher than that of the single-layer ring equalizer. Additionally, an experimental prototype of four batteries is established to verify the working principle and system equalization efficiency of the ring equalization circuit. The experimental results show that the equalization efficiency of the double-layer ring equalizer is 26.78% higher than that of single-layer ring equalizer. The simulation and experimental results are consistent with the theoretical analysis, which indicates that the circuit structure is reasonable and effective. [ABSTRACT FROM AUTHOR]

Can lifelong bilingualism be robustly decoded from intrinsic brain connectivity? Can we determine, using a spectrally resolved approach, the oscillatory networks that better predict dual‐language experience? We recorded resting‐state magnetoencephalographic activity in highly proficient Spanish‐Basque bilinguals and Spanish monolinguals, calculated functional connectivity at canonical frequency bands, and derived topological network properties using graph analysis. These features were fed into a machine learning classifier to establish how robustly they discriminated between the groups. The model showed excellent classification (AUC: 0.91 ± 0.12) between individuals in each group. The key drivers of classification were network strength in beta (15–30 Hz) and delta (2–4 Hz) rhythms. Further characterization of these networks revealed the involvement of temporal, cingulate, and fronto‐parietal hubs likely underpinning the language and default‐mode networks (DMNs). Complementary evidence from a correlation analysis showed that the top‐ranked features that better discriminated individuals during rest also explained interindividual variability in second language (L2) proficiency within bilinguals, further supporting the robustness of the machine learning model in capturing trait‐like markers of bilingualism. Overall, our results show that long‐term experience with an L2 can be "brain‐read" at a fine‐grained level from resting‐state oscillatory network organization, highlighting its pervasive impact, particularly within language and DMN networks. [ABSTRACT FROM AUTHOR]

Whole-brain connectome data characterize the connections among distributed neural populations as a set of edges in a large network, and neuroscience research aims to systematically investigate associations between brain connectome and clinical or experimental conditions as covariates. A covariate is often related to a number of edges connecting multiple brain areas in an organized structure. However, in practice, neither the covariate-related edges nor the structure is known. Therefore, the understanding of underlying neural mechanisms relies on statistical methods that are capable of simultaneously identifying covariate-related connections and recognizing their network topological structures. The task can be challenging because of false-positive noise and almost infinite possibilities of edges combining into subnetworks. To address these challenges, we propose a new statistical approach to handle multivariate edge variables as outcomes and output covariate-related subnetworks. We first study the graph properties of covariate-related subnetworks from a graph and combinatorics perspective and accordingly bridge the inference for individual connectome edges and covariate-related subnetworks. Next, we develop efficient algorithms to exact covariate-related subnetworks from the whole-brain connectome data with an |$\ell_0$| norm penalty. We validate the proposed methods based on an extensive simulation study, and we benchmark our performance against existing methods. Using our proposed method, we analyze two separate resting-state functional magnetic resonance imaging data sets for schizophrenia research and obtain highly replicable disease-related subnetworks. [ABSTRACT FROM AUTHOR]

Healthy aging is associated with a heterogeneous decline across cognitive functions, typically observed between language comprehension and language production (LP). Examining resting‐state fMRI and neuropsychological data from 628 healthy adults (age 18–88) from the CamCAN cohort, we performed state‐of‐the‐art graph theoretical analysis to uncover the neural mechanisms underlying this variability. At the cognitive level, our findings suggest that LP is not an isolated function but is modulated throughout the lifespan by the extent of inter‐cognitive synergy between semantic and domain‐general processes. At the cerebral level, we show that default mode network (DMN) suppression coupled with fronto‐parietal network (FPN) integration is the way for the brain to compensate for the effects of dedifferentiation at a minimal cost, efficiently mitigating the age‐related decline in LP. Relatedly, reduced DMN suppression in midlife could compromise the ability to manage the cost of FPN integration. This may prompt older adults to adopt a more cost‐efficient compensatory strategy that maintains global homeostasis at the expense of LP performances. Taken together, we propose that midlife represents a critical neurocognitive juncture that signifies the onset of LP decline, as older adults gradually lose control over semantic representations. We summarize our findings in a novel synergistic, economical, nonlinear, emergent, cognitive aging model, integrating connectomic and cognitive dimensions within a complex system perspective. [ABSTRACT FROM AUTHOR]

Network-group targets are a set of objectives that adhere to a shared communication protocol, perform common tasks, and exhibit relatively coordinated movements. Typically, network-group targets emit radar and communication signals. However, they often employ a silent mode to evade our perception. Despite this, they continue to exchange data through their communication networks. By intercepting the communication signals of these targets, this paper proposes a method for estimating the state and network topology of network-group targets based on random finite set (RFS) theory. This method is based on the modeling of network-group targets using a labeled multi-Bernoulli (LMB) RFS. In state estimation, the usual method involves extracting the localization parameters from the signals in the first step and use these parameters for target tracking in the second step. This study focused on estimating the kinematic states of network-group targets using communication signals containing delay and Doppler information received by multiple mobile sensor platforms. The proposed method considers the coherency between frequency-hopping pulses in the signals, resulting in an improved estimation performance. In addition, considering that network-group targets require network access for information exchange, graph theory concepts were utilized to estimate the network topology of network-group targets by address measurement. The simulation results validated the effectiveness of the proposed method and demonstrated its superior performance. [ABSTRACT FROM AUTHOR]

Skeleton‐based action recognition methods commonly employ graph neural networks to learn different aspects of skeleton topology information However, these methods often struggle to capture contextual information beyond the skeleton topology. To address this issue, a Scene Context‐aware Graph Convolutional Network (SCA‐GCN) that leverages potential contextual information in the scene is proposed. Specifically, SCA‐GCN learns the co‐occurrence probabilities of actions in specific scenarios from a common knowledge base and fuses these probabilities into the original skeleton topology decoder, producing more robust results. To demonstrate the effectiveness of SCA‐GCN, extensive experiments on four widely used datasets, that is, SBU, N‐UCLA, NTU RGB + D, and NTU RGB + D 120 are conducted. The experimental results show that SCA‐GCN surpasses existing methods, and its core idea can be extended to other methods with only some concatenation operations that consume less computational complexity. [ABSTRACT FROM AUTHOR]

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]

One of the well-known NP-hard optimization problems in graph theory is finding the longest path in a graph. This problem remains NP-hard for general grid graphs, and its complexity is open for grid graphs that have a limited number of holes. In this paper, we study this problem for odd-sized O -shaped grid graphs, i.e. a rectangular grid graph with a rectangular hole. We show that this problem can be solved in linear time. [ABSTRACT FROM AUTHOR]

Brain networks are highly optimized, connected systems across multiple spatiotemporal dimensions ranging from nano- to macroscales, which provide a balance between segregation of information and its integration. Stroke often involves a local area of brain damage followed by global functional and structural connectome perturbations, engaging brain networks at distinct spatial and temporal scales. Concurrent advances in brain readout and tracing technologies allow application of the conceptual framework of complex networks to decipher brain structural and functional reorganization after stroke. Application of multiscale connectomics-based approaches is necessary to delineate the underlying effect of stroke on brain structural and functional configuration and establish causal relationships. Insights gained through these approaches may lead to novel treatments for stroke. Stroke is a leading cause of adult disability. Understanding stroke damage and recovery requires deciphering changes in complex brain networks across different spatiotemporal scales. While recent developments in brain readout technologies and progress in complex network modeling have revolutionized current understanding of the effects of stroke on brain networks at a macroscale, reorganization of smaller scale brain networks remains incompletely understood. In this review, we use a conceptual framework of graph theory to define brain networks from nano- to macroscales. Highlighting stroke-related brain connectivity studies at multiple scales, we argue that multiscale connectomics-based approaches may provide new routes to better evaluate brain structural and functional remapping after stroke and during recovery. [ABSTRACT FROM AUTHOR]

The investigation into the domination problem and the associated subset problem of graphs is a focal point in graph theory research, sparking widespread interest and extensive exploration. This paper mainly studies the Italian domination of the strong product of two cycles. By constructing recursive Italian dominating functions, a well-defined bound for the Italian domination number in Cn Cm is obtained. Furthermore, through mathematical derivation and proof, the precise Italian domination number of C3 Cm is determined. [ABSTRACT FROM AUTHOR]

A chain graph is a bigraph where the neighborhood of vertices in each part forms a chain under set inclusion. Chain graphs have received considerable attention from researchers in the field of spectral graph theory, as they have the maximum spectral radius among all the bigraphs of prescribed size and order. Nevertheless, the areas of some graph parameters remain untouched. The reciprocal Wiener index or the Harary index is one of the distance-based topological indices among several graph parameters designed to capture the different aspects of molecular structure. This article explores the Harary index of chain graphs, giving the bounds and other properties. Further, the Harary index of threshold graphs, a slight structural variant of chain graphs is also discussed. The main focus is on chain graphs with integer-valued Harary index. The article presents a quadratic time algorithm for the inverse Harary index problem for chain graphs and hence contributes significantly to the theory of inverse problems on topological indices. [ABSTRACT FROM AUTHOR]

A critical component of information security is data encryption, and numerous researchers have invented various approaches to encrypt data. This paper presents a novel image encryption approach using the shortest weight first algorithm from graph theory, categorized as a symmetric cipher where the user has much freedom in selecting the keys. Furthermore, this method requires no sophisticated computations and is simple to understand and implement while providing effective protection against cryptographic attacks, such as brute force. The detailed algorithm, analysis, and security implications are also presented. The results firmly set a good base for many real-world applications involving security. [ABSTRACT FROM AUTHOR]