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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]

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]

We identify a sufficient condition, treewidth-pliability, that gives a polynomial-time algorithm for an arbitrarily good approximation of the optimal value in a large class of Max-2-CSPs parameterised by the class of allowed constraint graphs (with arbitrary constraints on an unbounded alphabet). Our result applies more generally to the maximum homomorphism problem between two rational-valued structures. The condition unifies the two main approaches for designing a polynomial-time approximation scheme. One is Baker's layering technique, which applies to sparse graphs such as planar or excluded-minor graphs. The other is based on Szemerédi's regularity lemma and applies to dense graphs. We extend the applicability of both techniques to new classes of Max-CSPs. However, we prove that the condition cannot be used to find solutions (as opposed to approximating the optimal value) in general. Treewidth-pliability turns out to be a robust notion that can be defined in several equivalent ways, including characterisations via size, treedepth, or the Hadwiger number. We show connections to the notions of fractional-treewidth-fragility from structural graph theory, hyperfiniteness from the area of property testing, and regularity partitions from the theory of dense graph limits. These may be of independent interest. In particular, we show that a monotone class of graphs is hyperfinite if and only if it is fractionally-treewidth-fragile and has bounded degree. [ABSTRACT FROM AUTHOR]

The article focuses on studying randomized greedy matching algorithms, particularly the Modified Randomized Greedy (MRG) algorithm and its weaker version named Random Decision Order (RDO). It mentions the RDO algorithm is shown to provide a 0.639-approximation for bipartite graphs and a 0.531-approximation for general graphs, leading to a substantial improvement in the approximation ratio of MRG.

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]

As industrial Internet of things (IIoT) for Manufacturing Service Collaboration (MSC) is becoming the current trend to accelerate the upgrade iteration of manufacturing capability, developing the robust IIoT platform operation mechanism for MSC is crucial to promote the continuous and stable service collaboration in the presence of supply and demand uncertainties. This paper studies the operational robustness of the IIoT platform for MSC. Firstly, the operation performances, requirements, and challenges of the IIoT platform towards manufacturing collaboration are analysed in classified platform practices, which can provide a comprehensive cognition about platform operation for manufacturing collaboration. Then, to evaluate the tolerance and persistence capabilities of MSC under supply and demand uncertainties, a graph-based operational robustness analysis method of the IIoT platform for MSC is proposed. The IIoT platform operation network for MSC is modelled as an interdependent network-of-network structure based on graph theory, which helps to characterise MSC performance properties under complexities. By combining manufacturing properties with network statistics, the evaluation metrics of operational robustness are established, which is done to quantise the MSC effectiveness under uncertainty effects. A case about customised manufacturing of automobiles illustrates the application of the proposed methods. Finally, future studies about robust MSC regulation are discussed. [ABSTRACT FROM AUTHOR]

Numerous domains, including all branches of science and technology, database theory, data mining, neural networks, expert systems, cluster analysis, control theory, and image capture, have employed graph theory, which can be used to represent interactions among multiple individuals. An extension of fuzzy sets, specifically intuitionistic fuzzy sets, is the concept of single-valued neutrosophic sets. We extend to graphs the notion of single-valued neutrosophic sets, which is an instance of neutrosophic sets. Neutrosophic fuzzy graphs are generalizations of fuzzy graphs, intuitionistic fuzzy graphs, and interval-valued fuzzy graphs. When in the case of fuzzy or intuitionistic fuzzy graphs relation between atoms in problems is indeterminate, then the neutrosophic fuzzy graph serves as a potent tool for dealing with partial, ambiguous, and inconsistent information in the actual world. In single-valued neutrosophic fuzzy graphs, each vertex is allocated a triplet in which the first element represents membership value, second and third elements as indeterminacy and non-membership value, respectively. In this work, we generalize the conclusions relating crisp graphs and fuzzy graphs by applying graph theory to single-valued neutrosophic sets and exploring a novel type of graph structure known as single-valued neutrosophic graphs. Sombor numbers are an effective and efficient tool in describing the topology of the graph in a numerical value. The main aim of this article is to define the first six Sombor numbers for single-valued neutrosophic fuzzy graphs. Then, degree of vertices of different graph families is determined in single-valued neutrosophic fuzzy framework and calculate the six Sombor numbers for these fundamental graph families for single-valued neutrosophic fuzzy graphs. In the end, an application to show the trade flows between different countries via sea route is presented to prove that single-valued neutrosophic fuzzy graphs are more generalized and efficient. First, we calculated the six Sombor numbers for crisp graph and then in SVNS fuzzy framework. Illustrated cases are then provided to show the applicability, viability, efficacy, and benefits of the suggested ways. [ABSTRACT FROM AUTHOR]

Ecological interactions among benthic communities are crucial for shaping marine ecosystems. Understanding these interactions is essential for predicting how ecosystems will respond to environmental changes, invasive species, and conservation management. However, determining the prevalence of species interactions at the community scale is challenging. To overcome this challenge, we employ tools from social network analysis, specifically exponential random graph modelling (ERGM). Our approach explores the relationships among animal and plant organisms within their neighbourhoods. Inspired by companion planting in agriculture, we use spatiotemporal co-occurrence as a measure of mixed species interaction. In other words, the variety of community interactions based on co-occurrence defines what we call 'co-occurrence social diversity'. Our objective is to use ERGM to quantify the proportion of interactions at both the simple paired level and the more complex triangle level, enabling us to measure and compare co-occurrence social diversity. Applying our approach to the Spanish coastal zone across eight sites, five depths, and sunlit/shaded aspects, we discover that 80% of sessile communities, consisting of over a hundred species, exhibit co-occurrence social diversity, with 5% of species consistently forming associations with other species. These organism-level interactions probably have a significant impact on the overall character of the site. This article is part of the theme issue 'Connected interactions: enriching food web research by spatial and social interactions'. [ABSTRACT FROM AUTHOR]

Ecological interactions among benthic communities are crucial for shaping marine ecosystems. Understanding these interactions is essential for predicting how ecosystems will respond to environmental changes, invasive species, and conservation management. However, determining the prevalence of species interactions at the community scale is challenging. To overcome this challenge, we employ tools from social network analysis, specifically exponential random graph modelling (ERGM). Our approach explores the relationships among animal and plant organisms within their neighbourhoods. Inspired by companion planting in agriculture, we use spatiotemporal co-occurrence as a measure of mixed species interaction. In other words, the variety of community interactions based on co-occurrence defines what we call 'co-occurrence social diversity'. Our objective is to use ERGM to quantify the proportion of interactions at both the simple paired level and the more complex triangle level, enabling us to measure and compare co-occurrence social diversity. Applying our approach to the Spanish coastal zone across eight sites, five depths, and sunlit/shaded aspects, we discover that 80% of sessile communities, consisting of over a hundred species, exhibit co-occurrence social diversity, with 5% of species consistently forming associations with other species. These organism-level interactions probably have a significant impact on the overall character of the site. This article is part of the theme issue 'Connected interactions: enriching food web research by spatial and social interactions'. [ABSTRACT FROM AUTHOR]

Spreading processes on networks (graphs) have become ubiquitous in modern society with prominent examples such as infections, rumors, excitations, contaminations, or disturbances. Finding the source of such processes based on observations is important and difficult. We abstract the problem mathematically as an optimization problem on graphs. For the deterministic setting we make connections to the metric dimension of a graph and introduce the concept of spread resolving sets. For the stochastic setting we propose a new algorithm combining parameter estimation and experimental design. We discuss well-posedness of the algorithm and show encouraging numerical results on a benchmark library. [ABSTRACT FROM AUTHOR]

This paper investigates a multi‐pursuer single‐evader nonzero‐sum differential game. Unlike traditional treatments, cooperative tracking is considered to design the optimal pursuit strategy by introducing a novel weighted topology and a scalar coupling gain. First, the distributed control strategy is designed based on the Hamilton–Jacobi–Isaacs (HJI) under the nonzero‐sum differential game. Then the modified LQR‐based cooperative tracking strategy is applied to achieve cooperativity among all the pursuers. The interception condition is also analyzed using a coupling gain based on the Laplacian matrix. Several illustrative examples under different scenarios are presented, and the results show that the pursuers with the proposed strategy can successfully intercept the evader. [ABSTRACT FROM AUTHOR]

This paper presents an edge‐based event‐triggered delay distributed algorithm for solving the economic dispatch problem (EDP) in smart grids. The objective of the EDP is to minimize the total generation cost by allocating power to individual generators, each with its local generation constraint. To save the overhead of communication resources between agents, an edge‐based event‐triggered mechanism is suggested. By setting distinct triggering thresholds for each communication edge, the agent may efficiently regulate the communication frequency among all neighbors. However, in practice, owing to the instability of the network, the agent may receive information regarding its neighbors, leading to communication lags for every communication link. The virtual agent technique and double‐stochastic matrix augmentation technique provide an equivalent delay‐free EDP. It is demonstrated that the proposed algorithm can asymptotically converge to the global optimal solution as long as the communication delay is arbitrary, time‐varying, and random, but bounded. Objective and clear evidence is provided through a case study in smart grids to verify the algorithm's feasibility. [ABSTRACT FROM AUTHOR]

Seismic source characterization is a crucial part of earthquake early warning. With the increasing seismic stations and collected data, some deep learning methods are gradually introduced and perform well in earthquake magnitude evaluation and localization. However, how to handle the sparse and non-European multi-stations is still a problem in earthquake multi-station models. This paper designs a multi-station model based on a graph neural network to accomplish seismic source characterization. The model applies the methods of graph theory to represent earthquake data as graph structure and innovatively adds the earthquake phase picks into the edges of the graph. This method mines the potential information among multi-stations effectively. The proposed methods improve the predicting precision and perform better in real-time performance than the compared models. [ABSTRACT FROM AUTHOR]

Let G be a connected graph and S be a k element subset of the vertex set V (G). The Steiner-k distance dG(S) between vertices of S is the minimum size among all connected subgraphs whose vertex set contains S. In this paper, we have defined the Steiner k-eccentric connectivity index and derived a closed formula for the same in case of some standard graphs. Also, we have used Steiner 3-eccentric connectivity index to predict values of boiling point of some primary and secondary amines, cross sectional area and molar refraction of alcohols. For each, regression model is developed and statistical analysis is conducted and these have ensured at least 97% accuracy. [ABSTRACT FROM AUTHOR]

In this paper, we define the zero-divisor graph of a meethyperlattice with respect to a hyperideal. We prove the diameter of a P-hyperlattice and Nakano hyperlattice are at most 3 and 4 respectively. We obtain that the zero-divisor graph with respect to the intersection of two prime hyperideals is complete bipartite. We prove certain properties of these zero-divisor graphs with suitable examples. [ABSTRACT FROM AUTHOR]

Purpose: Habitat connectivity is integral to current biodiversity science and conservation strategies. Originally, the connectivity concept stressed the role of individual movements for landscape-scale processes. Connectivity determines whether populations can survive in sub-optimal patches (i.e., source-sink effects), complete life cycles relying on different habitat types (i.e., landscape complementation), and benefit from supplementary resources distributed over the landscape (i.e., landscape supplementation). Although the past decades have witnessed major improvements in habitat connectivity modeling, most approaches have yet to consider the multiplicity of habitat types that a species can benefit from. Without doing so, connectivity analyses potentially fail to meet one of their fundamental purposes: revealing how complex individual movements lead to landscape-scale ecological processes. Methods: To bridge this conceptual and methodological gap, we propose to include multiple habitat types in spatial graph models of habitat connectivity, where nodes traditionally represent a single habitat type. Multiple habitat graphs will improve how we model connectivity and related landscape ecological processes, and how they are impacted by land cover changes. Results: In three case studies, we use these graphs to model (i) source-sink effects, (ii) landscape supplementation, and (iii) complementation processes, in urban ecosystems, agricultural landscapes, and amphibian habitat networks, respectively. A new version of the Graphab open-source software implements the proposed approach. Conclusion: Multiple habitat graphs help address crucial conservation challenges (e.g., urban sprawl, biological control, climate change) by representing more accurately the dynamics of populations, communities, and their interactions. Our approach thereby extends the ecologist’s toolbox and aims at fostering the alignment between landscape ecology theory and practice. [ABSTRACT FROM AUTHOR]

"Dogs" are connected to "cats" in our minds, and "backyard" to "outdoors." Does the structure of this semantic knowledge differ across people? Network-based approaches are a popular representational scheme for thinking about how relations between different concepts are organized. Recent research uses graph theoretic analyses to examine individual differences in semantic networks for simple concepts and how they relate to other higher-level cognitive processes, such as creativity. However, it remains ambiguous whether individual differences captured via network analyses reflect true differences in measures of the structure of semantic knowledge, or differences in how people strategically approach semantic relatedness tasks. To test this, we examine the reliability of local and global metrics of semantic networks for simple concepts across different semantic relatedness tasks. In four experiments, we find that both weighted and unweighted graph theoretic representations reliably capture individual differences in local measures of semantic networks (e.g., how related pot is to pan versus lion). In contrast, we find that metrics of global structural properties of semantic networks, such as the average clustering coefficient and shortest path length, are less robust across tasks and may not provide reliable individual difference measures of how people represent simple concepts. We discuss the implications of these results and offer recommendations for researchers who seek to apply graph theoretic analyses in the study of individual differences in semantic memory. [ABSTRACT FROM AUTHOR]

Mobility is a crucial element in comprehending the possible expansion of the transmission chain in an epidemic. In the initial phases, strategies for containing cases can be directly linked to population mobility restrictions, especially when only non-pharmaceutical measures are available. During the pandemic of COVID-19 in Brazil, mobility limitation measures were strongly opposed by a large portion of the population. Hypothetically, if the population had supported such measures, the sharp rise in the number of cases could have been suppressed. In this context, computational modeling offers systematic methods for analyzing scenarios about the development of the epidemiological situation taking into account specific conditions. In this study, we examine the impacts of interstate mobility in Brazil. To do so, we develop a metapopulational model that considers both intra and intercompartmental dynamics, utilizing graph theory. We use a parameter estimation technique that allows us to infer the effective reproduction number in each state and estimate the time-varying transmission rate. This makes it possible to investigate scenarios related to mobility and quantify the effect of people moving between states and how certain measures to limit movement might reduce the impact of the pandemic. Our results demonstrate a clear association between the number of cases and mobility, which is heightened when states are closer to each other. This serves as a proof of concept and shows how reducing mobility in more heavily trafficked areas can be more effective. [ABSTRACT FROM AUTHOR]

Graph theory is used in many areas of chemical sciences, especially in molecular chemistry. It is particularly useful in the structural analysis of chemical compounds and in modeling chemical reactions. One of its applications concerns determining the structural formula of a chemical compound. This can be modeled as a variant of the well-known graph realization problem. In the classical version of the problem, a sequence of natural numbers is given, and the question is whether there exists a graph in which the vertices have degrees equal to the given numbers. In the variant considered in this paper, instead of a sequence of natural numbers, a sequence of sets of natural numbers is given, and the question is whether there exists a multigraph such that each of its vertices has a degree equal to a number from one of the sets. This variant of the graph realization problem matches the nature of the problem of determining the structural formula of a chemical compound better than other variants considered in the literature. We propose a polynomial time exact algorithm solving this variant of the problem. [ABSTRACT FROM AUTHOR]

Background: Essential tremor (ET) and Parkinson's disease (PD) are the two most prevalent movement disorders, sharing several overlapping tremor clinical features. Although growing evidence pointed out that changes in similar brain network nodes are associated with these two diseases, the brain network topological properties are still not very clear. Objective: The combination of graph theory analysis with machine learning (ML) algorithms provides a promising way to reveal the topological pathogenesis in ET and tremor-dominant PD (tPD). Methods: Topological metrics were extracted from Resting-state functional images of 86 ET patients, 86 tPD patients, and 86 age- and sex-matched healthy controls (HCs). Three steps were conducted to feature dimensionality reduction and four frequently used classifiers were adopted to discriminate ET, tPD, and HCs. Results: A support vector machine classifier achieved the best classification performance of four classifiers for discriminating ET, tPD, and HCs with 89.0% mean accuracy (mACC) and was used for binary classification. Particularly, the binary classification performances among ET vs. tPD, ET vs. HCs, and tPD vs. HCs were with 94.2% mACC, 86.0% mACC, and 86.3% mACC, respectively. The most power discriminative features were mainly located in the default, frontal-parietal, cingulo-opercular, sensorimotor, and cerebellum networks. Correlation analysis results showed that 2 topological features negatively and 1 positively correlated with clinical characteristics. Conclusions: These results demonstrated that combining topological metrics with ML algorithms could not only achieve high classification accuracy for discrimination ET, tPD, and HCs but also help to reveal the potential brain topological network pathogenesis in ET and tPD. [ABSTRACT FROM AUTHOR]

In this paper, we introduce the subset-strong product of graphs and give a method for calculating the adjacency spectrum of this product. In addition, exact expressions for the first and second Zagreb indices of the subset-strong products of two graphs are reported. Examples are provided to illustrate the applications of this product in some growing graphs and complex networks. [ABSTRACT FROM AUTHOR]

Water distribution networks, like other large infrastructure systems, must consider reliability and resilience efforts to resist and recover from failure with limited resources. Management of these assets requires a plan to prioritize the maintenance of system components, which are key to its reliability. Many current asset management practices for water distribution networks include only reactive strategies, such as fixing components after breaking, or on predetermined maintenance schedules, which might not correlate with a component's condition or need. Asset management and graph theory principles are utilized in this study to evaluate a water distribution network and its vulnerabilities at Tyndall Air Force Base (AFB). We begin by estimating pipe condition indices and establishing the importance of each pipe through a set of graph-theoretic measures (e.g., pipe betweenness centrality and network efficiency). We next simulate two pipe failure scenarios: (1) a single pipe failure scenario, and (2) a cascading failure scenario. The outcomes of the second scenario reveal that a larger betweenness centrality equates to being a critical pipe in relation to overall system performance. Further, we estimate the risk associated with each pipe by combining pipe condition and network properties. We find that ∼7 km (or ∼6%) of pipes are at high risk, while the majority of the network's pipes (∼79 km or ∼62%) are low risk. This study demonstrates how integrating condition indices with graph theory metrics could be employed to promote best practices in pipe maintenance and create a risk-informed framework for asset management. [ABSTRACT FROM AUTHOR]

Evaluating the architect's role in construction contracts is vital for improving the contractual relationship and preventing disputes. Standard contracts define the architect as an owner's agent and/or a contract administrator. Studies often use qualitative and subjective evaluations to analyze the architect's role. However, there is a lack of quantitative measures that objectively evaluate the architect's capacities in construction contracts. This paper fills this knowledge gap. This study proposes a quantitative graph theory-based approach to analyze the architect's roles under three standard forms of construction contracts in the US: (1) extracting and classifying party interactions; (2) modeling the interactions using network analysis; (3) conducting an expert-based survey to determine the importance of the interactions; and (4) developing the architect's-capacity score (ACS) as an index that depicts the architect's roles into one interpretable measure. The calculated ACS established the American Institute of Architects (AIA) at the upper end of the spectrum followed by the Engineers Joint Contract Documents Committee contract and then finally the ConsensusDocs with the lowest score. Such delineates the architect's presence under the AIA contract as both an owner's agent as well as a contract's administrator. The EJCDC contract processes reflected the architect's role to be more of an owner's agent. Finally, the ConsensusDocs represented the lowest score due to the lack of the architect's presence in most contract processes. This study contributes to the body of knowledge by providing a methodological approach that mediates the subjectivity entailed in the evaluation of the architect's role: one quantifiable measure. Similarly, Utilizing ACS enables project parties to customize the architect's involvement in the construction phase through tailored contractual arrangements. This quantification of the architect's role promotes a clearer understanding of the architect's expectations, fostering enhanced communication and trust. [ABSTRACT FROM AUTHOR]

Given a graph L $L$, the Turán number ex(n,L) $\text{ex}(n,L)$ is the maximum possible number of edges in an n $n$‐vertex L $L$‐free graph. The study of Turán number of graphs is a central topic in extremal graph theory. Although the celebrated Erdős‐Stone‐Simonovits theorem gives the asymptotic value of ex(n,L) $\text{ex}(n,L)$ for nonbipartite L $L$, it is challenging in general to determine the exact value of ex(n,L) $\text{ex}(n,L)$ for χ(L)≥3 $\chi (L)\ge 3$. The odd‐ballooning of H $H$ is a graph such that each edge of H $H$ is replaced by an odd cycle and all new vertices of odd cycles are distinct. Here the length of odd cycles is not necessarily equal. The exact value of Turán number of the odd‐ballooning of H $H$ is previously known for H $H$ being a cycle, a path, a tree with assumptions, and K2,3 ${K}_{2,3}$. In this paper, we manage to obtain the exact value of Turán number of the odd‐ballooning of Ks,t ${K}_{s,t}$ with 2≤s≤t $2\le s\le t$, where (s,t)∉{(2,2),(2,3)} $(s,t)\notin \{(2,2),(2,3)\}$ and each odd cycle has length at least five. [ABSTRACT FROM AUTHOR]

In classical graph theory, the minimal spanning tree (MST) is a subgraph that lacks cycles and efficiently connects every vertex by utilizing edges with the minimum weights. The computation of a minimum spanning tree for a graph has been a pervasive problem over time. However, in practical scenarios, uncertainty often arises in the form of fuzzy edge weights, leading to the emergence of the Fuzzy Minimum Spanning Tree (FMST). This specialized approach is adept at managing the inherent uncertainty present in edge weights within a fuzzy graph, a situation commonly encountered in real-world applications. This study introduces the initial optimization approach for the Minimum Spanning Tree Problem within the context of interval-valued fermatean neutrosophic domain. The proposed solution involves the adaptation of the Dhouib-Matrix-MSTP (DM-MSTP) method, an innovative technique designed for optimal resolution. The DM-MSTP method operates by employing a column-row navigation strategy through the adjacency matrix. To the best of our knowledge, instances of this specific problem have not been addressed previously. To address this gap, a case study is generated, providing a comprehensive application of the novel DM-MSTP method with detailed insights into its functionality and efficacy. [ABSTRACT FROM AUTHOR]

Objective: The alterations of the functional network (FN) in anti-N-methyl-Daspartate receptor (NMDAR) encephalitis have been recognized by functional magnetic resonance imaging studies. However, few studies using the electroencephalogram (EEG) have been performed to explore the possible FN changes in anti-NMDAR encephalitis. In this study, the aim was to explore any FN changes in patients with anti-NMDAR encephalitis. Methods: Twenty-nine anti-NMDAR encephalitis patients and 29 age- and gender-matched healthy controls (HC) were assessed using 19-channel EEG examination. For each participant, five 10-second epochs of resting state EEG with eyes closed were extracted. The cortical source signals of 84 Brodmann areas were calculated using the exact low resolution brain electromagnetic tomography (eLORETA) inverse solution by LORETA-KEY. Phase Lag Index (PLI) matrices were then obtained and graph and relative band power (RBP) analyses were performed. Results: Compared with healthy controls, functional connectivity (FC) in the delta, theta, beta 1 and beta 2 bands significantly increased within the 84 cortical source signals of anti-NMDAR encephalitis patients (p < 0.05) and scalp FC in the alpha band decreased within the 19 electrodes. Additionally, the anti-NMDAR encephalitis group exhibited higher local efficiency and clustering coefficient compared to the healthy control group in the four bands. The slowing band RBP increased while the fast band RBP decreased in multiple-lobes and some of these changes in RBP were correlated with the modified Rankin Scale (mRS) and Mini-mental State Examination (MMSE) in anti-NMDAR encephalitis patients. Conclusions: This study further deepens the understanding of related changes in the abnormal brain network and power spectrum of anti-NMDA receptor encephalitis. The decreased scalp alpha FC may indicate brain dysfunction, while the increased source beta FC may indicate a compensatory mechanism for brain function in anti-NMDAR encephalitis patients. These findings extend understanding of how the brain FN changes from a cortical source perspective. Further studies are needed to detect correlations between altered FNs and clinical features and characterize their potential value for the management of anti-NMDAR encephalitis. [ABSTRACT FROM AUTHOR]

Topological indices are mathematical descriptors of the structure of a molecule that can be used to predict its properties. They are derived from the graph theory, which describes the topology of a molecule and its connectivity. The main objective is mathematical modeling and topological properties of Y-graphyne. Current research focuses on two structures made from hexagonal honeycomb graphite lattices named triangular Y-graphyne and triangular Y-graphyne chains. The authors have simultaneously computed the first and second Reverse Zagreb indices, reverse hyper-Zagreb indices, and their polynomials. This research also derives mathematical closed-form formulas for some of its fundamental degree-based molecular descriptors. Researchers have been trying to synthesize a novel carbon form called Graphyne. For over a decade but with no success. Recently, some researchers have made a breakthrough in generating Carbons elusive allotrope and solved a long-standing problem in carbon materials. This wonder material is created to rival the conductivity of graphene but with control. These results opened new ways of research in the fields of semiconductors, electronics and optics. Furthermore, graphical and tabular results will help to investigate the structure-property relationships in γ-graphyne. [ABSTRACT FROM AUTHOR]

The hypothalamus in the mammalian brain is responsible for regulating functions associated with survival and reproduction representing a complex set of highly interconnected, yet anatomically and functionally distinct, sub-regions. It remains unclear what factors drive the spatial organization of sub-regions within the hypothalamus. One potential factor may be structural connectivity of the network that promotes efficient function with well-connected sub-regions placed closer together geometrically, i.e., the strongest axonal signal transferred through the shortest geometrical distance. To empirically test for such efficiency, we use hypothalamic data derived from the Allen Mouse Brain Connectivity Atlas, which provides a structural connectivity map of mouse brain regions derived from a series of viral tracing experiments. Using both cost function minimization and comparison with a weighted, sphere-packing ensemble, we demonstrate that the sum of the distances between hypothalamic sub-regions are not close to the minimum possible distance, consistent with prior whole brain studies. However, if such distances are weighted by the inverse of the magnitude of the connectivity, their sum is among the lowest possible values. Specifically, the hypothalamus appears within the top 94th percentile of neural efficiencies of randomly packed configurations and within one standard deviation of the median efficiency when packings are optimized for maximal neural efficiency. Our results, therefore, indicate that a combination of geometrical and topological constraints help govern the structure of the hypothalamus. [ABSTRACT FROM AUTHOR]

A general method for the calculation of the flow and pressure of a gas in a network of cylindrical capillaries is presented. This method is used specifically for gas chromatographic systems in this work. With this approach, it is possible to easily calculate flow and pressures in complex gas chromatographic systems, like flow‐modulated or thermal‐modulated multidimensional gas chromatographic systems, or systems with multiple outlets at different pressures. A mathematic abstraction using graph theory is used to represent the system of capillaries. With this graph, the flow balance equations at the connections of the capillaries can easily be set up. Using a computer algebra system, the system of flow balance equations can be solved for the pressures at the connection points. For simple systems, this approach is presented, and calculated flows, pressures, and hold‐up times are compared with measured values. In addition, two complex systems (4‐Way‐Splitter, Deans Switch system) of capillaries are presented with calculations only. For these systems, certain conditions were formulated, that is, a certain difference in hold‐up times and a defined split ratio between different paths of these systems. Using a numeric non‐linear solver, configurations of these systems were found, that fulfill these conditions. [ABSTRACT FROM AUTHOR]

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]

The scale and cluster development of offshore wind power faces the constraints of high construction cost, limited submarine corridor resources, etc. Under the background of offshore wind power parity development, it is urgent to consider the actual constraints of the project and the impact of project income on the owner, and the construction of offshore public station and the optimal selection of transmission method have become an important measure to reduce the cost and increase the efficiency of offshore wind power. A life-cycle cost model for offshore wind farm cluster grid-connection system planning is constructed to consider the project income, and the offshore public station location and capacity, transmission method, and high-voltage cable selection and path are optimized, which are solved by an improved single parent genetic algorithm and a topology repair strategy based on graph theory. The results show that the proposed planning scheme can effectively improve the life-cycle project income of the grid-connection system and provide technical support for the grid-connection planning of large-scale offshore wind farm clusters. [ABSTRACT FROM AUTHOR]

Recent work has exposed the idea that interesting quantum-like (QL) probability laws, including interference effects, can be manifest in classical systems. Here, we propose a model for QL states and QL bits. We suggest a way that huge, complex systems can host robust states that can process information in a QL fashion. Axioms that such states should satisfy are proposed. Specifically, it is shown that building blocks suited for QL states are networks, possibly very complex, that we defined based on k -regular random graphs. These networks can dynamically encode a lot of information that is distilled into the emergent states we can use for QL processing. Although the emergent states are classical, they have properties analogous to quantum states. Concrete examples of how QL functions are possible are given. The possibility of a 'QL advantage' for computing-type operations and the potential relevance for new kinds of function in the brain are discussed and left as open questions. [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]

Green spaces and blue spaces in cities provide a wealth of benefits to the urban social–ecological system. Unfortunately, urban development fragments natural habitats, reducing connectivity and biodiversity. Urban green–blue infrastructure (UGI) networks can mitigate these effects by providing ecological corridors that enhance habitat connectivity. This study examined UGI connectivity for two indicator species in a rapidly developing city in the southern United States. We mapped and analyzed UGI at a high resolution (0.6 m) across the entire city, with a focus on semi-natural areas in private land and residential neighborhoods. Integrating graph theory and a gravity model, we assessed structural UGI networks and ranked them based on their ability to support functional connectivity. Most of the potential habitat corridors we mapped in this project traversed private lands, including 58% of the priority habitat for the Golden-cheeked Warbler and 69% of the priority habitat for the Rio Grande Wild Turkey. Riparian zones and other areas with dense tree cover were critical linkages in these habitat corridors. Our findings illustrate the important role that private semi-natural areas play in UGI, habitat connectivity, and essential ecosystem services. [ABSTRACT FROM AUTHOR]

Suicide is a major concern for health, and depression is an established proximal risk factor for suicide. This study aimed to investigate white matter features associated with suicide. We constructed white matter structural networks by deterministic tractography via diffusion tensor imaging in 51 healthy controls, 47 depressed patients without suicide plans or attempts and 56 depressed patients with suicide plans or attempts. Then, graph theory analysis was used to measure global and nodal network properties. We found that local efficiency was decreased and path length was increased in suicidal depressed patients compared to healthy controls and non-suicidal depressed patients; moreover, the clustering coefficient was decreased in depressed patients compared to healthy controls; and the global efficiency and normalized characteristic path length was increased in suicidal depressed patients compared to healthy controls. Similarly, compared with those in non-suicidal depressed patients, nodal efficiency in the thalamus, caudate, medial orbitofrontal cortex, hippocampus, olfactory cortex, supplementary motor area and Rolandic operculum was decreased. In summary, compared with those of non-suicidal depressed patients, the structural connectome of suicidal depressed patients exhibited weakened integration and segregation and decreased nodal efficiency in the fronto-limbic-basal ganglia-thalamic circuitry. These alterations in the structural networks of depressed suicidal brains provide insights into the underlying neurobiology of brain features associated with suicide. [ABSTRACT FROM AUTHOR]

Previous studies have shown that language acquisition influences both the structure and function of the brain. However, whether the acquisition of a second language at different periods of life alters functional network organization in different ways remains unclear. Here, functional magnetic resonance imaging data from 27 English-speaking monolingual controls and 52 Spanish–English bilingual individuals, including 22 early bilinguals who began learning a second language before the age of ten and 30 late bilinguals who started learning a second language at age fourteen or later, were collected from the OpenNeuro database. Topological metrics of resting-state functional networks, including small-world attributes, network efficiency, and rich- and diverse-club regions, that characterize functional integration and segregation of the networks were computed via a graph theoretical approach. The results showed obvious increases in network efficiency in early bilinguals and late bilinguals relative to the monolingual controls; for example, the global efficiency of late bilinguals and early bilinguals was improved relative to that of monolingual controls, and the local efficiency of early bilinguals occupied an intermediate position between that of late bilinguals and monolingual controls. Obvious increases in rich-club and diverse-club functional connectivity were observed in the bilinguals relative to the monolingual controls. Three network metrics were positively correlated with Spanish proficiency test scores. These findings demonstrated that early and late acquisition of a second language had different impacts on the functional networks of the brain. [ABSTRACT FROM AUTHOR]

This study introduces the notion of complex t-neutrosophic graphs (CTNGs) as a powerful tool for understanding and displaying complex interactions that are sometimes difficult to understand. It demonstrates that CTNGs may accurately reflect complicated interactions involving several components or dimensions within a particular scenario. It also instructs the basic set operations of CTNGs and analyzes notions like homomorphism and isomorphism within this framework. Furthermore, the research describes a practical application of CTNGs. It illustrates their value in addressing biodiversity conservation by taking into account a variety of relevant factors. The paper uses this application to highlight the flexibility and effectiveness of CTNGs as a tool for decision-makers to visualize and prioritize activities targeted at improving biodiversity conservation. [ABSTRACT FROM AUTHOR]

The clique transversal problem is a critical concept in graph theory, focused on identifying a minimum subset of vertices that intersects all maximal cliques in a graph. This problem and its variations—such as the k-fold clique, { k } -clique, minus clique, and signed clique transversal problems—have received significant interest due to their theoretical importance and practical applications. This paper examines the k-fold clique, { k } -clique, minus clique, and signed clique transversal problems on distance-hereditary graphs. Known for their distinctive structural properties, distance hereditary graphs provide an ideal framework for studying these problem variants. By exploring these issues in the context of distance-hereditary graphs, this research enhances the understanding of the computational challenges and the potential for developing efficient algorithms to address these problems. [ABSTRACT FROM AUTHOR]

In this study, we explore the main supergraph S (G) of a finite group G , defined as an undirected, simple graph with a vertex set G in which two distinct vertices, a and b , are adjacent in S (G) if the order of one is a divisor of the order of the other. This is denoted as either o (a) ∣ o (b) or o (b) ∣ o (a) , where o (⋅) is the order of an element. We classify finite groups for which the main supergraph is either a split graph or a threshold graph. Additionally, we characterize finite groups whose main supergraph is a cograph. Our classification extends to finite groups G with S (G) , a cograph that includes when G is a direct product of two non-trivial groups, as well as when G is either a dihedral group, a generalized quaternion group, a symmetric group, an alternating group, or a sporadic simple group. [ABSTRACT FROM AUTHOR]

This paper extends the framework of soft set theory to directed graphs by introducing and analyzing the concepts of the normal product and restricted normal product of soft directed graphs. Building on Molodtsov’s foundational work in developing soft set theory to address uncertainty in data, this study presents new methods for modeling and understanding complex systems where uncertainty plays a significant role. Soft directed graphs, which enhance traditional graph models by incorporating parameters and uncertain relationships, serve as the foundation for this investigation. The normal product, defined as a combination of two soft directed graphs based on their respective parameter sets, and the restricted normal product, which combines soft directed graphs only where their parameter sets intersect, provide a comprehensive framework for these new operations. This paper also establishes the structural properties of these products, ensuring they are well-defined and retain the key features of soft directed graphs. Furthermore, we derive combinatorial identities related to vertex and arc counts, as well as degree sums, offering deeper insights into the composition and behavior of these graph products. [ABSTRACT FROM AUTHOR]

Sierpiński graphs are frequently related to fractals, and fractals apply in several fields of science, i.e., in chemical graph theory, computer networking, biology, and physical sciences. Functions and polynomials are powerful tools in computer mathematics for predicting the features of networks. Topological descriptors, frequently graph constraints, are absolute values that characterize the topology of a computer network. In this essay, Firstly, we compute the M-polynomials for Sierpiński-type fractals. We derive some degree-dependent topological invariants after applying algebraic operations on these M-polynomials. [ABSTRACT FROM AUTHOR]

The use of biofuels represents a promising means of achieving a sustainable future and offers considerable economic and environmental benefits. Since they are derived from organic sources, such as vegetable oils and animal fats, biofuels can mitigate the effects of greenhouse gas emissions, improve air quality, support local agriculture, create employment opportunities, and enhance energy security by reducing dependence on fossil fuels. However, introducing these alternative fuels to the aviation sector remains a significant challenge. Thus, it is vital to investigate the potential of sustainable aviation fuel (SAF) and discover how to overcome the technological obstacles to its integration into mainstream aviation to attain broader decarbonization objectives. This article seeks to contribute to a discussion about SAF by examining how it has evolved and its connections to related patents. This article is a comprehensive study of biofuel innovation, highlighting the complex relationships between academia, industry, and other stakeholders. It is hoped that the findings from this study will provide a clearer understanding of the catalysts involved in SAF innovation and provide valuable insights for policymakers, academics, and professionals in the field who are committed to shaping the trajectory of sustainable energy technologies in the future. [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]

Background: The alternations of brain responses to a strong desire to void were unclear, and the gender differences under the strong desire to void remain controversial. The present study aims to identify the functional brain network's topologic property changes evoked by a strong desire to void in healthy male and female adults with synchronous urodynamics using a graph theory analysis. Methods: The bladders of eleven healthy males and eleven females were filled via a catheter using a specific infusion and withdrawal pattern. A resting-state functional magnetic resonance imaging (fMRI) was performed on the enrolled subjects, scanning under both the empty bladder and strong desire to void states. An automated anatomical labeling (AAL) atlas was used to identify the ninety cortical and subcortical regions. Pearson's correlation calculations were performed to establish a brain connection matrix. A paired t-test (p < 0.05) and Bonferroni correction were applied to identify the significant statistical differences in topological properties between the two states, including small-world network property parameters [gamma (γ) and lambda (λ)], characteristic path length (Lp), clustering coefficient (Cp), global efficiency (Eglob), local efficiency (Eloc), and regional nodal efficiency (Enodal). Results: The final data suggested that females and males had different brain response patterns to a strong desire to void, compared with an empty bladder state. Conclusions: More brain regions involving emotion, cognition, and social work were active in females, and males might obtain a better urinary continence via a compensatory mechanism. [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 crisp graph theory, there are numerous topological indices accessible, including the Misbalance Prodeg Index, which is one of the most well-known degree-based topological indexes. In crisp graphs, both vertices and edges have membership values of 1 or 0, whereas in fuzzy graphs, both vertices and edges have different memberships. This is an entire contrast to the crisp graph. In this paper, we introduce the Fuzzy Misbalance Prodeg Index of a fuzzy graph, which is a generalized form of the Misbalance Prodeg Index of a graph. We find bounds of this index and find bounds of certain classes of graphs such as path graph, cycle graph, complete graph, complete bipartite graph, and star graph. We give an algorithm to find the Fuzzy Misbalance Prodeg Index of a graph for the model of multi-criteria decision-making is established. We present applications from daily life in multi-criteria decision-making. We apply our obtained model of the Fuzzy Misbalance Prodeg Index for the multicriteria decision-making to the choice of the best supplier and we also show the graphical analysis of our index with the other indices that show how our index is better than other existing indices. [ABSTRACT FROM AUTHOR]