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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]

Background and aimsRecently, we demonstrated that a distinct pattern of structural covariance networks (SCN) from magnetic resonance imaging (MRI)-derived measurements of brain cortical thickness characterized young adults with alcohol use disorder (AUD) and predicted current and future problematic drinking in adolescents relative to controls. Here, we establish the robustness and value of SCN for identifying heavy alcohol users in three additional independent studies.Design and settingCross-sectional and longitudinal studies using data from the Pediatric Imaging, Neurocognition and Genetics (PING) study (n = 400, age range = 14-22 years), the National Consortium on Alcohol and Neurodevelopment in Adolescence (NCANDA) (n = 272, age range = 17-22 years) and the Human Connectome Project (HCP) (n = 375, age range = 22-37 years).CasesCases were defined based on heavy alcohol use patterns or former alcohol use disorder (AUD) diagnoses: 50, 68 and 61 cases were identified. Controls had none or low alcohol use or absence of AUD: 350, 204 and 314 controls were selected.MeasurementsGraph theory metrics of segregation and integration were used to summarize SCN.FindingsMirroring our prior findings, and across the three data sets, cases had a lower clustering coefficient [area under the curve (AUC) = -0.029, P = 0.002], lower modularity (AUC = -0.14, P = 0.004), lower average shortest path length (AUC = -0.078, P = 0.017) and higher global efficiency (AUC = 0.007, P = 0.010). Local efficiency differences were marginal (AUC = -0.017, P = 0.052). That is, cases exhibited lower network segregation and higher integration, suggesting that adjacent nodes (i.e. brain regions) were less similar in thickness whereas spatially distant nodes were more similar.ConclusionStructural covariance network (SCN) differences in the brain appear to constitute an early marker of heavy alcohol use in three new data sets and, more generally, demonstrate the utility of SCN-derived metrics to detect brain-related psychopathology.

Passive vibration absorbers (PVAs) play a crucial role in mitigating excessive vibrations in engineering structures. Traditional PVA design typically begins with proposing a beneficial topological layout, incorporating stiffness, damping, and inertance elements, followed by optimal sizing of each element to minimise specific response of dynamically excited structures. An alternative approach involves first designing the impedance function of a PVA and then identifying a passive mechanical layout that replicates this impedance using network synthesis techniques. However, both methods struggle to identify the most efficient PVA layout using the minimum number of elements (referred to as "complexity") for a given vibration suppression problem. To this end, this study introduces a graph‐based methodology for designing optimal configurations (i.e., layout + sizing) of two‐terminal spring‐damper‐inerter PVAs that achieve specified performance goals with minimum complexity. In this approach, a PVA is represented as a weighted coloured multigraph, enabling the application of a novel graph‐based enumeration technique to generate the full set of potential layouts from any given number of mechanical elements. This enumeration is followed by a performance assessment of all layouts to pinpoint the optimal absorber configuration for the given problem. The methodology's automation capability and versatility make it suitable for various civil and mechanical engineering applications. The effectiveness of the proposed methodology is demonstrated through two case studies: a vibration absorber design for a wind‐excited tall building and a suspension design for a road vehicle. In both cases, the proposed methodology successfully identifies innovative PVA layouts that surpass traditional designs with minimum additional elements. [ABSTRACT FROM AUTHOR]

ABSTRACT A nonlinear variational auto‐encoder (NLVAE) is developed to reconstruct the plane strain stress field in a solid with embedded cracks subjected to uniaxial tension, uniaxial compression, and shear loading paths. Latent features are sampled from a skew‐normal distribution, which allows encoding marked variations of the features of the stress field across the load steps. The NLVAE is trained and tested based upon stress maps generated with the finite element method (FEM) with cohesive zone elements (CZEs). The NLVAE successfully captures stress concentrations that develop across the loading steps as a result of crack propagation, especially when enhanced disentanglement is emphasized during training. Some latent variables consistently emerge as significant across various microstructure descriptors and loading paths. Correlations observed between the evolution of fabric descriptors and that of their significant stress latent features indicate that the NLVAE can capture important microstructure transitions during the loading process. Crack connectivity, crack eccentricity, and the distribution of zones of highly connected opened cracks versus zones with no cracks are the fabric descriptors that best explain the sequences of latent features that are the most important for the reconstruction of the stress field. Notably, the distributional shape, tail behavior, and symmetry of microstructure descriptor distributions have more influence on the stress field than basic measures of central tendency and spread. [ABSTRACT FROM AUTHOR]

This paper presents Guidance and Control (G&C) systems for multi-satellite assembly in proximity operations. The systems utilize the twistor model, which is linearized through Taylor's series. Decentralized control laws, designed using Linear Quadratic Regulator (LQR) and Model Predictive Control (MPC), are employed to track an energy-optimal trajectory generated using the Hamiltonian approach. Data exchange between satellites and their neighbors is represented using graph theory. The decentralized MPC framework is implemented using the CasADi package. To ensure collision avoidance between the satellites, a repulsive control law is designed, considering symmetric input saturation in the actuators. The proposed G&C systems are tested using a high-fidelity nonlinear satellite relative motion model that incorporates orbital perturbations. Numerical simulations are performed in a MATLAB® environment, and the results are visualized using STK®. Furthermore, a comparative study is conducted to evaluate tracking performance and fuel consumption between the two control methods. The results demonstrate that the use of an optimal trajectory reduces fuel consumption for both control algorithms. [ABSTRACT FROM AUTHOR]

A classical result in spectral graph theory states that if a graph G has an equitable partition π then the eigenvalues of the divisor graph G π are a subset of its eigenvalues, i.e. σ (G π) ⊆ σ (G). A natural question is whether it is possible to recover the remaining eigenvalues σ (G) − σ (G π) in a similar manner. Here we show that any weighted undirected graph with nontrivial equitable partition can be decomposed into a number of subgraphs whose collective spectra contain these remaining eigenvalues. Using this decomposition, which we refer to as a complete equitable decomposition, we introduce an algorithm for finding the eigenvalues of an undirected graph (symmetric matrix) with a nontrivial equitable partition. Under mild assumptions on this equitable partition we show that we can find eigenvalues of such a graph faster using this method when compared to standard methods. This is potentially useful as many real-world data sets are quite large and have a nontrivial equitable partition. [ABSTRACT FROM AUTHOR]

We prove the inequality λ k ≤ d k + d k − 1 for all the eigenvalues λ 1 ≤ λ 2 ≤ ⋯ ≤ λ n of the Kirchhoff matrix K of a finite simple graph or quiver with vertex degrees d 1 ≤ d 2 ≤ ⋯ ≤ d n and assuming d 0 = 0. Without multiple connections, the inequality λ k ≥ max (0 , d k − (n − k)) holds. A consequence in the finite simple graph or multi-graph case is that the pseudo determinant Det (K) counting the number of rooted spanning trees has an upper bound 2 n ∏ k = 1 n d k and that det (1 + K) counting the number of rooted spanning forests has an upper bound ∏ k = 1 n (1 + 2 d k). [ABSTRACT FROM AUTHOR]

Diagnosability is a fundamental consideration when designing an interconnected network. The PMC and MM ∗ fault diagnosis models are the two most commonly used models. Both the g -good-neighbour diagnosability and g -extra diagnosability of an interconnection network have been two of the hot topics in the intersectional research areas of Graph theory and Computer Science, which become increasingly attractive for new solutions to real-world problems. However, there are still some problems in the transformation from the concepts of Computer Science to that of mathematics. In this paper, we systematically study such problems and give a strict proof from concepts to mathematical conclusions. In the terms of results, we not only give the relationship between g -good-neighbour diagnosabilities of the network under PMC model and MM ∗ model, but also between g -extra diagnosabilities of the network under PMC and MM ∗ models. To apply our results, we give an application on the enhanced hypercube in the end and derive a lemma explaining whether these are 3-cycles in enhanced hypercubes and how many common neighbours for two vertices of enhanced hypercubes under different values of k in the meantime. [ABSTRACT FROM AUTHOR]

The strongest genetic risk factor for Alzheimer's disease (AD) is the ε4 allele of apolipoprotein E (ApoE ε4). A high fat diet also adds to the risk of dementia and AD. In addition, there are sex differences as women carriers have a higher risk of an earlier onset and rapid decline in memory than men. The present study looked at the effect of the genetic risk of ApoE ε4 together with a high fat/high sucrose diet (HFD/HSD) on brain function in male and female rats using magnetic resonance imaging. We hypothesized female carriers would present with deficits in cognitive behavior together with changes in functional connectivity as compared to male carriers. Four-month-old wildtype and human ApoE ε4 knock-in (TGRA8960), male and female Sprague Dawley rats were put on a HFD/HSD for four months. Afterwards they were imaged for changes in function using resting state BOLD functional connectivity. Images were registered to, and analyzed, using a 3D MRI rat atlas providing site-specific data on 173 different brain areas. Resting state functional connectivity showed male wildtype had greater connectivity between areas involved in feeding and metabolism while there were no differences between female and male carriers and wildtype females. The data were unexpected. The genetic risk was overshadowed by the diet. Male wildtype rats were most sensitive to the HFD/HSD presenting with a deficit in cognitive performance with enhanced functional connectivity in neural circuitry associated with food consumption and metabolism. [ABSTRACT FROM AUTHOR]

Graph theory enables a direct quantification of topological properties of any arbitrary network. Its application in neuroscience has unveiled topological changes of brain networks associated with various neurodegenerative diseases. This study used the graph theory to understand speech deficits in patients with Parkinson's disease (PD). In particular, this study investigated the effect of subthalamic nucleus deep brain stimulation (STN-DBS) on the topology of speech graphs. Sixty patients with PD completed a standard semantic fluency test with DBS switched ON and OFF. A control group of sixty matched nonsurgical PD patients completed the test once. All verbal responses were recorded, transcripted, and transformed into directed speech graphs. Volumes of tissue activated (VTA) were estimated for three STN subregions, including sensorimotor, associative, and limbic parts. First, the patients with DBS OFF produced smaller and denser speech graphs than nonsurgical patients, showing fewer nodes, higher density, shorter diameter, and shorter average shortest path. Second, DBS partially reversed the effect of surgery, leading to larger and sparser speech graphs with more nodes, lower density, longer diameter, and longer average shortest path (ON versus OFF). Third, however, the left associative VTA negatively correlated with the DBS-induced diameter and average shortest path changes (ON versus OFF), suggesting that the patients with greater left associative STN stimulation tended to produce smaller and denser speech graphs. This study demonstrates that STN-DBS can partially restore the topological structure of speech graphs in PD patients. However, stimulating the left associative STN appears to disrupt speech graphs. [ABSTRACT FROM AUTHOR]

AbstractIn graph theory, a graph is a mathematical structure that consists of a set of vertices and a set of edges connecting those pairs of vertices. Let G=(V,E) be a graph, a dominating set for G is a subset D⊆V of vertices such that every vertex in V∖D has at least one neighbor in D. The domination number of G, denoted by γ(G), is the minimum cardinality of a dominating set in G. Recently, Khan (2023) performed a comparative study of seven important domination parameters with the π-electronic energy of benzenoid hydrocarbons, with a new theoretical approach. Physico-chemical properties are crucial for understanding the physical and chemical behavior of a molecule. In this paper, we perform a comparative study of those seven domination parameters with the physico-chemical properties of benzenoid hydrocarbons. We find the values of those domination parameters for the 22 lower benzenoid hydrocarbons and follow the statistical approach described by Khan (2023). The two physico-chemical properties to be chosen are the normal boiling point Tb, a representative for intermolecular and van-der-Waals type interactions, whereas, the standard enthalpy/heat of formation ΔHfo is referred to represent the thermal properties of a molecular graph. This study further enhances that the performance of paired domination number is exceptional among all other parameters and highly correlated with both of the physico-chemical properties. Its correlation coefficient is 0.9981 (respectively, 0.9726) with Tb (respectively, ΔHfo). The outstanding correlation coefficient of paired domination number ensures further usage in quantitative structure-property relationship (QSPR) models. [ABSTRACT FROM AUTHOR]

In the current age of chemical science, chemical graph theory has significantly advanced our understanding of the characteristics of chemical compounds. To simulate the mathematical, chemical, and physical aspects of networks, a topological index, a numerical measure obtained from the graph of a chemical network, employed. Recent work has explored the topological properties of boron oxide using chemical graph theory. In this work, we conduct a Pearson correlation analysis of boron oxide to assess the correlations between the Van and S indices and entropy metrics. We analyze the Pearson correlation coefficients between the entropy values and the calculated indices using a heatmap. In this article, a significant positive correlation between the Van, and S indices, and entropy values, which is represented by the heatmap of the strong linear correlations. To avoid duplication, a dimensionality reduction technique should be used for highly connected variables. Additionally, this study gives a detailed explanation of the link between the indices and entropy, which will form the basis of further statistical investigations. [ABSTRACT FROM AUTHOR]

The preface to "Geometry and Topology with Applications" highlights the historical significance and evolution of geometry as a research field in mathematics. It discusses the contributions of prominent mathematicians like Euclid, Euler, Gauss, Lobachevsky, Riemann, Hilbert, Poincaré, Thurston, and Gromov to the development of various geometries and topologies. The document also focuses on the work of Thurston and Gromov in geometric topology and geometric group theory, respectively, and their impact on modern mathematics. The Special Issue attracted submissions from researchers worldwide, with 12 papers accepted for publication, including survey papers on discrete groups and topology in the fourth dimension. [Extracted from the article]

In recent years, phase linking (PL) methods in radar time-series interferometry (TSI) have proven to be powerful tools in geodesy and remote sensing, enabling the precise monitoring of surface displacement and deformation. While these methods are typically designed to operate on a complete network of interferograms, generating such networks is often challenging in practice. For instance, in non-urban or vegetated regions, decorrelation effects lead to significant noise in long-term interferograms, which can degrade the time-series results if included. Additionally, practical issues such as gaps in satellite data, poor acquisitions, or systematic errors during interferogram generation can result in incomplete networks. Furthermore, pre-existing interferogram networks, such as those provided by systems like COMET-LiCSAR, often prioritize short temporal baselines due to the vast volume of data generated by satellites like Sentinel-1. As a result, complete interferogram networks may not always be available. Given these challenges, it is critical to understand the applicability of PL methods on these incomplete networks. This study evaluated the performance of two PL methods, eigenvalue decomposition (EVD) and eigendecomposition-based maximum-likelihood estimator of interferometric phase (EMI), under various network configurations including short temporal baselines, randomly sparsified networks, and networks where low-coherence interferograms have been removed. Using two sets of simulated data, the impact of different network structures on the accuracy and quality of the results was assessed. These patterns were then applied to real data for further comparison and analysis. The findings demonstrate that while both methods can be effectively used on short temporal baselines, their performance is highly sensitive to network sparsity and the noise introduced by low-coherence interferograms, requiring careful parameter tuning to achieve optimal results across different study areas. [ABSTRACT FROM AUTHOR]

Though youth living in disadvantaged neighborhoods experience greater risk for poor behavioral and mental health outcomes, many go on to show resilience in the face of adversity. A few recent studies have identified neural markers of resilience in cognitive and affective brain networks, yet the broader network organization supporting resilience in youth remains unknown, particularly in relation to neighborhood disadvantage. Moreover, most studies have defined resilience as the absence of psychopathology, which does not consider growing evidence that resilience also includes positive outcomes across multiple domains (e.g., social, academic). We examined associations between brain network organization and multiple resilience domains in a sample of 708 twins (7–19 years old) recruited from neighborhoods with above-average poverty levels. Graph analysis on functional connectivity data from resting-state and task-based functional magnetic resonance imaging was used to characterize features of intrinsic whole-brain and network-level organization, from which we explored associations with resilience in three domains: psychological, social, and academic. Fewer connections between a brain network involved in self-referential processing (i.e., default mode network) and the subcortical system were associated with greater social resilience. Further, greater whole-brain functional integration (i.e., efficiency) was associated with better psychological resilience among youth with relatively lower levels of cumulative adversity exposure. Alternatively, lower whole-brain efficiency and higher whole-brain robustness to disruption (i.e., assortativity) were associated with greater psychological and social resilience among youth with relatively higher levels of cumulative adversity. These findings advance support for multidimensional resilience models and reveal distinct neural mechanisms supporting resilience to neighborhood disadvantage across specific domains in youth. Public Significance Statement: The present study used brain imaging to examine how the organization of information flow throughout the brain relates to resilience to neighborhood disadvantage in youth. Results identified three domains of resilience (i.e., psychological, social, and academic), which showed unique associations with patterns of information flow across brain networks and the entire brain. This work bridges advancements in resilience theory with neuroscience to identify brain markers of resilience among youth marginalized by the socioeconomic status of their neighborhood. [ABSTRACT FROM AUTHOR]

Purpose: This review highlights the importance of functional connectivity in pediatric neuroscience, focusing on its role in understanding neurodevelopment and potential applications in clinical practice. It discusses various techniques for analyzing brain connectivity and their implications for clinical interventions in neurodevelopmental disorders. Methods: The principles and applications of independent component analysis and seed-based connectivity analysis in pediatric brain studies are outlined. Additionally, the use of graph analysis to enhance understanding of network organization and topology is reviewed, providing a comprehensive overview of connectivity methods across developmental stages, from fetuses to adolescents. Results: Findings from the reviewed studies reveal that functional connectivity research has uncovered significant insights into the early formation of brain circuits in fetuses and neonates, particularly the prenatal origins of cognitive and sensory systems. Longitudinal research across childhood and adolescence demonstrates dynamic changes in brain connectivity, identifying critical periods of development and maturation that are essential for understanding neurodevelopmental trajectories and disorders. Conclusion: Functional connectivity methods are crucial for advancing pediatric neuroscience. Techniques such as independent component analysis, seed-based connectivity analysis, and graph analysis offer valuable perspectives on brain development, creating new opportunities for early diagnosis and targeted interventions in neurodevelopmental disorders, thereby paving the way for personalized therapeutic strategies. [ABSTRACT FROM AUTHOR]

Freezing of gait (FOG) is a disabling motor symptom prevalent in patients with Parkinson's disease (PD); however, its pathophysiological mechanisms are poorly understood. This study aimed to investigate whole‐brain functional connectivity (FC) pattern alterations in PD patients with FOG. A total of 18 PD patients, 10 with FOG (PD‐FOG) and 8 without FOG (PD‐nFOG), and 10 healthy controls were enrolled. High‐resolution 3D T1‐weighted and resting‐state functional MRI (rs‐fMRI) data were obtained from all participants. The groups' internetwork connectivity differences were explored with rs‐fMRI FC using seed‐based analysis and graph theory. Multiple linear regression analysis estimated the relationship between FC changes and clinical measurements. Rs‐fMRI analysis demonstrated alterations in FC in various brain regions between the three groups. Freezing of Gait Questionnaire severity was correlated with decreased brain functional connection between Vermis12 and the left temporal occipital fusiform cortex (r = −0.82, P <.001). Graph theory topological metrics indicated a decreased clustering coefficient in the right superior temporal gyrus in the PD‐nFOG group. PD‐FOG patients exhibited a compensatory increase in connectivity between the left inferior frontal gyrus language network and the postcentral gyrus compared to PD‐nFOG patients. Further, the decreased connection between Vermis 12 and the left temporal occipital fusiform cortex may serve as a potential neuroimaging biomarker for tracking PD‐FOG and distinguishing between PD subtypes. [ABSTRACT FROM AUTHOR]

Mirror symmetry for higher genus curves is usually formulated and studied in terms of Landau–Ginzburg models; however the critical locus of the superpotential is arguably of greater intrinsic relevance to mirror symmetry than the whole Landau–Ginzburg model. Accordingly, we propose a new approach to the A-model of the mirror, viewed as a trivalent configuration of rational curves together with some extra data at the nodal points. In this context, we introduce a version of Lagrangian Floer theory and the Fukaya category for trivalent graphs, and show that homological mirror symmetry holds, namely, that the Fukaya category of a trivalent configuration of rational curves is equivalent to the derived category of a non-Archimedean generalized Tate curve. To illustrate the concrete nature of this equivalence, we show how explicit formulas for theta functions and for the canonical map of the curve arise naturally under mirror symmetry. [ABSTRACT FROM AUTHOR]

For k , ℓ ∈ N , we introduce the concepts of 푘-ultrahomogeneity and ℓ-tuple regularity for finite groups. Inspired by analogous concepts in graph theory, these form a natural generalization of homogeneity, which was studied by Cherlin and Felgner as well as Li, and automorphism transitivity, which was investigated by Zhang. Additionally, these groups have an interesting algorithmic interpretation. We classify the 푘-ultrahomogeneous and ℓ-tuple regular finite groups for k , ℓ ≥ 2 . In particular, we show that every 2-tuple regular finite group is ultrahomogeneous. [ABSTRACT FROM AUTHOR]

In this research, medications used for treating heart disease, specifically focusing on calcium channel blockers, were analyzed. A computer-based computing technique was used to simplify calculations and data analysis. Using MATLAB coding, we calculated their degree-based topological indices obtained from the M-polynomial. Various regression analyses were used to establish a relationship between these indices and the physicochemical features of the drugs. QSPR models were created to determine the effectiveness by correlating these indices with eight physicochemical features of the drugs. Confidence intervals were calculated at a 95% level for the intercept and slope in the linear regression models. The results indicate that the inverse sum indeg index (I) proved to be the most dependable indices for predicting boiling point, flashpoint, and enthalpy of vaporization. The symmetric division index (SDD) was effective in forecasting polarizability and molar refractivity, while the second modified Zagreb index ( m M 2 ) emerged as the best predictor for molar volume in linear, quadratic, and cubic regression models. Furthermore, the forgotten index (F) was identified as the top estimator for boiling point, flashpoint, enthalpy, and polar surface area in both quadratic and cubic regression models. Lastly, the SDD index, with a correlation coefficient of R = 1, is proposed as the most accurate estimator for the characteristics of polar surface area in quadratic, and cubic regression equations. Calculated feature values show a strong correlation with the actual values, indicating the indices' reliable predictive capabilities. [ABSTRACT FROM AUTHOR]

A topological index is a numerical representation of a chemical network in chemical graph theory. Similar to their chemical representation, a topological descriptor identifies specific the physical attributes of the underlying chemical compounds. We describe the third type of hex-derived networks rectangular hex-derived network R H D N 3 , chain hex-derived network C H D N 3 , in our work (r). In this paper, modified and efficient formulas based on multiplicative graph invariants will be constructed. Such as ordinary geometric-arithmetic (OGA), general version of harmonic index (GHI), sum connectivity indes (SI), general sum connectivity index (GSI), first and second Gourava and hyper- Gourava indices, Shegehalli and Kanabur indices, first generalised version of Zagreb index (GZI) and forgotten index (FI) for the subdivided hex -derived R H D N 3 (r) , C H D N 3 (r) networks. In order to develop some new formulas we the compute the multiplicative topological properties. This study found several types of edges for computing and also discussed the order and size.To properly appreciate the chemical compounds, physical and biological characteristics we can combine the study of the networks described above with the chemical compounds and their graphical structures. These observations have the potential to evaluate bio and physiological activities. [ABSTRACT FROM AUTHOR]

Investigation of chemical applicability and mathematical properties of topological indices is the contemporary research trend in chemical graph theory. The geometric arithmetic index (GA) stands out as an well nurtured index having strong correlation with properties and activities of molecules. Its exponential variant (EGA) is investigated here from chemical and mathematical point of view. The chemical connection of EGA is explored by assessing its structure–property modelling ability and isomer discrimination potential. Its mathematical features are demonstrated by finding bounds for numerous classes of graphs with characterizing extremal structures. Second maximum and minimum trees are characterized with respect to the EGA index. [ABSTRACT FROM AUTHOR]

Being a core tool, graph theory, as a mathematical formalism in mathematical chemistry, has become an essential approach for studying the complex behavior and interactions in protein systems. Here, we review recent advances in the field, particularly in our group, including the methods developed to access protein functions and their applications in disease biology. First, we provide the necessary background and definitions of graph-based structures and network centralities, and methodologies developed at the node-, subgraph- and pathway-levels. We then review the applications of how to use these algorithms to gain new biological insights, ranging from protein structures to protein dynamics, and interactions for linking genotypes and phenotypes. Furthermore, we discuss immediate challenges in the multilayer network, which is more realistic in the biological world, and hope to draw increasing attention from mathematicians, especially graph theorists, to reveal the basic principles of "networks of networks". [ABSTRACT FROM AUTHOR]

For a graph G (V , E) and ℓ ∈ ℕ , an ℓ distance coloring is a coloring f : V → { 1 , 2 , ... , t } of V with t colors such that ∀ u , v ∈ V , u ≠ v , f (u) ≠ f (v) when d (u , v) ≤ ℓ. Here d (u , v) is the distance between u and v and is equal to the minimum number of edges that connect u and v in G. The span of ℓ distance coloring of G , χ ℓ (G) , is the minimum t among all ℓ distance coloring of G. A class of channel assignment problem in cellular network can be formulated as a distance graph coloring problem in regular grid graphs. The cellular network is often modelled as an infinite hexagonal grid H , and hence determining χ ℓ (H) has relevance from practical point of view. Jacko and Jendrol [Discussiones Mathematicae Graph Theory, 2005] determined the exact value of χ ℓ (H) for any odd ℓ and for even ℓ ≥ 8 , it is conjectured that χ ℓ (H) = 3 8 (ℓ + 4 3) 2 where [ x ] is an integer, x ∈ ℝ and x − 1 2 < [ x ] ≤ x + 1 2 . For ℓ = 8 , the conjecture has been proved by Ghosh and Koley [ 2 2 nd Italian Conference on Theoretical Computer Science, 2021]. In this paper, we prove the conjecture for any even ℓ ≥ 1 0. [ABSTRACT FROM AUTHOR]

The exploration of large-scale unknown environments can benefit from the deployment of multiple robots for collaborative mapping. Each robot explores a section of the environment and communicates onboard pose estimates and maps to a central server to build an optimized global multi-robot map. Naturally, inconsistencies can arise between onboard and server estimates due to onboard odometry drift, failures, or degeneracies. The mapping server can correct and overcome such failure cases using computationally expensive operations such as inter-robot loop closure detection and multi-modal mapping. However, the individual robots do not benefit from the collaborative map if the mapping server provides no feedback. Although server updates from the multi-robot map can greatly alleviate the robotic mission strategically, most existing work lacks them, due to their associated computational and bandwidth-related costs. Motivated by this challenge, this paper proposes a novel collaborative mapping framework that enables global mapping consistency among robots and the mapping server. In particular, we propose graph spectral analysis, at different spatial scales, to detect structural differences between robot and server graphs, and to generate necessary constraints for the individual robot pose graphs. Our approach specifically finds the nodes that correspond to the drift's origin rather than the nodes where the error becomes too large. We thoroughly analyze and validate our proposed framework using several real-world multi-robot field deployments where we show improvements of the onboard system up to 90% and can recover the onboard estimation from localization failures and even from the degeneracies within its estimation. [ABSTRACT FROM AUTHOR]

An ideal I of Q [ x 1 , ⋯ , x n ] is said to be m-absorbing if any monomial of total degree p > m belongs to I and if there is a monomial M of total degree m such that M ∉ I. Inspired by the fundamental work of Lechuga and Murillo (Topology 39:89–94, 2000) who established a connection between graph theory and rational homotopy theory, these paper aims to characterise the m-absorbing ideals of Q [ x 1 , ⋯ , x n ] generated by a family of complete homogeneous symmetric polynomials of the form P rs (k) = ∑ t = 1 k x r k - t x s t - 1 , where k ≥ 3. [ABSTRACT FROM AUTHOR]

We establish a general condition on the cost function to obtain uniqueness and Monge solutions in the multi-marginal optimal transport problem, under the assumption that a given collection of the marginals are absolutely continuous with respect to local coordinates. When only the first marginal is assumed to be absolutely continuous, our condition is equivalent to the twist on splitting sets condition found in Kim and Pass (SIAM J Math Anal 46:1538–1550, 2014; SIAM J Math Anal 46:1538–1550, 2014). In addition, it is satisfied by the special cost functions in our earlier work (Pass and Vargas-Jiménez in SIAM J Math Anal 53:4386–4400, 2021; Monge solutions and uniqueness in multi-marginal optimal transport via graph theory. arXiv:2104.09488, 2021), when absolute continuity is imposed on certain other collections of marginals. We also present several new examples of cost functions which violate the twist on splitting sets condition but satisfy the new condition introduced here, including a class of examples arising in robust risk management problems; we therefore obtain Monge solution and uniqueness results for these cost functions, under regularity conditions on an appropriate subset of the marginals. [ABSTRACT FROM AUTHOR]

Evolutionary graph theory has considerably advanced the process of modelling the evolution of structured populations, which models the interactions between individuals as pairwise contests. In recent years, these classical evolution models have been extended to incorporate more realistic features, e.g. multiplayer games. A recent series of papers have developed a new evolutionary framework including structure, multiplayer interactions, evolutionary dynamics, and movement. However, so far, the developed models have mainly considered independent movement without coordinated behaviour. Although the theory underlying the framework has been developed and explored in various directions, several movement mechanisms have been produced which characterise coordinated movement, for example, herding. By embedding these newly constructed movement distributions, within the evolutionary setting of the framework, we demonstrate that certain levels of aggregation and dispersal benefit specific types of individuals. Moreover, by extending existing parameters within the framework, we are not only able to develop a general process of embedding any of the considered movement distributions into the evolutionary setting on complete graphs but also analytically produce the probability of fixation of a mutant on a complete N-sized network, for the multiplayer Public Goods and Hawk–Dove games. Also, by applying weak selection methods, we extended existing previous analyses on the pairwise Hawk–Dove Game to encompass the multiplayer version considered in this paper. By producing neutrality and equilibrium conditions, we show that hawks generally do worse in our models due to the multiplayer nature of the interactions. [ABSTRACT FROM AUTHOR]

Smart city networks are critical for delivering essential services such as healthcare, education, and business operations. However, these networks are highly susceptible to a range of threats, including natural disasters and intentional cyberattacks, which can severely disrupt their functionality. To address these vulnerabilities, we present the resilient smart city (ResiSC) system, designed to enhance the resilience of smart city communication networks through a topological design approach. Our system employs a graph‐theoretic algorithm to determine the optimal network topology for a given set of nodes, aiming to maximize connectivity while minimizing link provisioning costs. We introduce two novel connectivity measurements, All Nodes Reachability (ANR) and Sum of All Nodes Reachability (SANR), to evaluate network resilience. We applied our approach to data from two public universities of different sizes, simulating various attack scenarios to assess the robustness of the resulting network topologies. Evaluation results indicate that our solution improves network resilience against targeted attacks by 38% compared to baseline methods such as k‐nearest neighbours (k‐NN) graphs, while also reducing the number of additional links and their associated costs. Results also indicate that our proposed solution outperforms baseline methods like k‐NN in terms of network resilience against targeted attacks by 41%. This work provides a practical framework for developing robust smart city networks capable of withstanding diverse threats. [ABSTRACT FROM AUTHOR]

A graph G is (2 , m) -linked if, for any distinct vertices a 1 , ... , a m , b 1 , b 2 in G , there exist disjoint connected subgraphs A , B of G such that a 1 , ... , a m ∈ V (A) and b 1 , b 2 ∈ V (B). A fundamental result in structural graph theory is the characterization of (2 , 2) -linked graphs. It appears to be difficult to characterize (2 , m) -linked graphs for m ≥ 3. In this paper, we provide a partial characterization of (2 , m) -linked graphs. This implies that every (2 m + 2) -connected graphs G is (2 , m) -linked and for any distinct vertices a 1 , ... , a m , b 1 , b 2 of G , there is a path P in G between b 1 and b 2 and avoiding { a 1 , ... , a m } such that G − P is connected, improving a previous connectivity bound of 10 m. [ABSTRACT FROM AUTHOR]

Soft set theory, introduced by D. Molodtsov, is a mathematical approach that tackles uncertain data. Semigraph is a generalization of the concept of a graph. By incorporating soft sets into semigraphs, a new concept called soft semigraphs has emerged. The study of soft semigraphs has gained significance in graph theory due to their valuable applications in parametrization. This paper explores the properties of various vertex degrees in soft semigraphs, including degrees, end degrees, edge degrees, adjacent degrees, and consecutive adjacent degrees. [ABSTRACT FROM AUTHOR]

Purpose: To determine whether individuals with subjective cognitive decline (SCD) have changes in whole-brain network characteristics and intracerebral node characteristics in the structural network, and whether there is a difference between SCD with and without Apolipoprotein E4 (APOEε4). Methods: This cross-sectional study included 36 individuals without SCD without APOEε4 (healthy control, HC group), 21 individuals with SCD with APOEε4 (APOEε4+ group), and 33 individuals with SCD without APOEε4 (APOEε4-group). The white matter structural network was constructed using the fractional anisotropy (FA) based deterministic fiber tracking method. Graph theory was used to analyze the whole-brain network characteristics and intracerebral node characteristics of the three groups. Results: Regarding the whole-brain network characteristics, all three groups exhibited small-worldness in their structural networks. The clustering coefficient (Cp) and local efficiency (Eloc) in the APOEε4+ and APOEε4- groups were significantly lower than in the HC group (p < 0.05), but no significant difference in Cp or Eloc was observed between the APOEε4+ and APOEε4- groups. Regarding intracerebral node characteristics, there were significant differences in some brain regions, mainly the default mode network (DMN), the occipital lobe, the temporal lobe, and subcortical regions. The change in intracerebral node characteristics was different between the APOEε4+ group and the APOEε4- group. Conclusions: Individuals with SCD demonstrate changes in whole-brain network characteristics and intracerebral node characteristics in the structural network. Moreover, differences exist between APOEε4+ and APOEε4- individuals. [ABSTRACT FROM AUTHOR]

Let S be a groupoid (magma) with zero 0, and let R = ⊕ s ∈ S R s be a contracted S-graded ring, that is, an S-graded ring with R 0 = 0. By G (H R) we denote the undirected power graph of a multiplicative subsemigroup H R = ∪ s ∈ S R s of R, and by G * (H R) a graph obtained from G (H R) by removing 0 and its incident edges. If Re is a nonzero ring component of R, then G * (R e) denotes a subgraph of G * (H R) , induced by R e *. In this paper we address a problem raised in [Abawajy, J., Kelarev, A., Chowdhury, M.: Power Graphs: A Survey. Electron. J. Graph Theory Appl. 1(2), 125–147 (2013)]. Namely, let S be torsion-free, that is, s n = t n implies s = t for all s, t ∈ S , and all positive integers n, and let S be 0-cancellative, that is, for all s, t, u ∈ S , su = tu ≠ 0 implies s = t , and us = ut ≠ 0 implies s = t. Also, let R be semisimple Artinian. We prove that if G * (R e) is connected for every nonzero ring component Re of R, then the connected components of G * (H R) are precisely the graphs G * (R e). [ABSTRACT FROM AUTHOR]

The issue of opinion sharing and formation has received considerable attention in the academic literature, and a few models have been proposed to study this problem. However, existing models are limited to the interactions among nearest neighbors, with those second, third, and higher-order neighbors only considered indirectly, despite the fact that higher-order interactions occur frequently in real social networks. In this paper, we develop a new model for opinion dynamics by incorporating long-range interactions based on higher-order random walks that can explicitly tune the degree of influence of higher-order neighbor interactions. We prove that the model converges to a fixed opinion vector, which may differ greatly from those models without higher-order interactions. Since direct computation of the equilibrium opinion is computationally expensive, which involves the operations of huge-scale matrix multiplication and inversion, we design a theoretically convergence-guaranteed estimation algorithm that approximates the equilibrium opinion vector nearly linearly in both space and time with respect to the number of edges in the graph. We conduct extensive experiments on various social networks, demonstrating that the new algorithm is both highly efficient and effective. [ABSTRACT FROM AUTHOR]

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This study contributes to Action, Process, Object, Schema (APOS) theory research by showing two approaches used by advanced mathematics students to construct relations between higher-order derivatives to solve complex problems. We show evidence of students' ability to perform Actions on their graphing derivative Schema, that is, of its thematization. It also contributes to the literature on the learning of differential calculus by showing how advanced students use their knowledge to construct relations between concepts when facing complex situations. The work of three graduate students on transforming complex graphs and determining their properties and their relation to the domain structure is analyzed to determine their solution approaches. Their graphing derivative Schema is analyzed in depth in terms of the construction of relations among the Schema structures and assimilation and accommodation mechanisms involved in thematization in APOS theory. These findings are important in informing and developing didactic strategies to foster university students' understanding of derivatives, which can smoothe the transition to the study of advanced mathematics courses. [ABSTRACT FROM AUTHOR]

Precipitation channelled down tree stems (stemflow) or into drip points of 'throughfall' beneath trees results in spatially concentrated inputs of water and chemicals to the ground. Currently, these flows are poorly characterised due to uncertainties about which branches redirect rainfall to stemflow or throughfall drip points.We introduce a graph theoretic algorithm that 'prunes' quantitative structural models of trees (derived from terrestrial LiDAR) to identify branches contributing to stemflow and those contributing to throughfall drip points. To demonstrate the method's utility, we analysed two trees with similar canopy sizes but contrasting canopy architecture and rainfall partitioning behaviours.For both trees, the branch 'watershed' area contributing to stemflow (under conditions assumed to represent moderate precipitation intensity) was found to be only half of the total ground area covered by the canopy. The study also revealed significant variations between trees in the number and median contribution areas of modelled throughfall drip points (69 vs. 94 drip points tree−1, with contributing projected areas of 28.6 vs. 7.8 m2 tree−1, respectively). Branch diameter, surface area, volumes and woody area index of components contributing to stemflow and throughfall drip points may play a role in the trees' differing rainfall partitioning behaviours.Our pruning algorithm, enabled by the proliferation of LiDAR observations of canopy structure, promises to enhance studies of canopy hydrology. It offers a novel approach to refine our understanding of how trees interact with rainfall, thereby broadening the utility of existing LiDAR data in environmental research. [ABSTRACT FROM AUTHOR]

The performance of the photoelectric tracking system mainly depends on the tracking accuracy. In order to achieve the purpose of high precision tracking, controlling the power dragging device, the main component of the photoelectric tracking system, is the main means to achieve this purpose. In order to improve the tracking speed and accuracy of photoelectric tracking systems, the author proposed a mathematical nonlinear graph theory topology layer model for photoelectric tracking systems. The topology layer model and the motion node servo mechanism model of the photoelectric tracking system are studied, and the two-stage disturbance sources that affect the tracking stability are analyzed. The results show that the interference estimation error range designed by the author reaches 3×10-3 through comparison, it can be seen that the estimation error designed by the author is significantly smaller than the ESO estimation error, and the buffeting is small. Through comparison, it can be concluded that the estimation error of the algorithm and the disturbance observer designed by the author are significantly smaller than the estimation error of ESO, and the buffeting is small, and they have strong compensation ability for the disturbance of different frequencies. The interference observer can quickly and accurately estimate the interference and prove its stability. The effectiveness of the proposed observer and disturbance observer is fully demonstrated. The finite time integrated site selection mode disturbance observer (F-ISMDOB) is designed to quickly estimate and compensate for equivalent interference, and effectively improves the anti-interference performance of the system structure layer. [ABSTRACT FROM AUTHOR]

This study presents a novel approach that employs a mixture of the tunable-Q wavelet transform (TQWT) and enhanced AdaBoost to address the issue of high impedance fault (HIF) recognition in power distribution networks. Traditional overcurrent protection relays frequently have lower fault current levels than normal current, making it exceedingly difficult to detect this HIF problem with the necessity to use a quick and effective approach to find HIF problems. Since the TQWT performs better with signals that exhibit oscillatory behavior, it has been utilized to extract special features for the training of the improved AdaBoost model. The procedure is accelerated by calculating the Kourtosis (K) value for each level and selecting the ideal level of decomposition to minimize computing work. Faulted zones are categorized using an enhanced AdaBoost approach. Under normal, noisy, and unbalanced conditions, the recommended approach is applied to an imbalanced 123-bus test system and an IEEE 33-bus test system. The efficiency of the recommended method is also being assessed for imbalanced distribution networks incorporating dispersed generation into real-time platforms. This procedure is quick compared to previous methods since it uses an upgraded AdaBoost classifier and optimal decomposition level. [ABSTRACT FROM AUTHOR]

In this article, the (p, q)–Stancu–Schurer–Bleimann–Butzer–Hahn ((p, q)‐SSBBH) operators are introduced. The Korovkin‐type theorem is obtained to show the approximation properties of these operators. Then, the rate of convergence of these operators with the help of the modulus of continuity and Lipschitz‐type maximal functions is calculated, respectively. Finally, for the asymptotic behavior of these operators, the Voronovskaja‐type theorem is given. Furthermore, the convergence of these operators to the considered function f by plotting the graphs is demonstrated. And, this convergence is compared with the convergence of the (p, q)–Bleimann–Butzer–Hahn ((p, q)‐BBH) operators to the same function. [ABSTRACT FROM AUTHOR]

A topological index is a numerical quantity for the structure graph of the molecule and it can be represented through graph theory. Also, its application not only in the field of chemistry can also be applied in areas including computer science, networking, etc. Hence, this paper introduces the Randic index of a vague graph and vague subgraph with their properties. The upper and lower boundaries of the Randic index of vague graphs are studied with some isomorphic properties. Likewise, the Randic index of directed vague graphs is introduced. Finally, an application of the Randic index in construction has been presented. [ABSTRACT FROM AUTHOR]

Sport-related concussion (SRC) is a complex brain injury. By applying graph-theoretical analysis to networks derived from neuroimaging techniques, studies have shown that despite an overall retention of small-world topology, changes in small-world properties occur after brain injury. Less is known about how exercise during athletes' return to sport (RTS) influences these brain network properties. Therefore, in the present study dense electroencephalography (EEG) datasets were collected pre- and post-moderate aerobic exercise. Small-world properties of whole brain (WB) and the default mode network (DMN) were extracted from the EEG datasets of 21 concussed athletes and 21 healthy matched controls. More specifically, path length (LP), clustering coefficient (CP), and small-world index (SWI) in binary and weighted graphs were calculated in the alpha frequency band (7–13 Hz). Pre-exercise, SRC athletes had higher DMN-CP values compared to controls, while post-exercise SRC athletes had higher WB-LP compared to controls. Weighted WB analysis revealed a significant association between SRC and the absence of small-world topology (SWI ≤ 1) post-exercise. This explorative study provides preliminary evidence that moderate aerobic exercise during athletes' RTS induces an altered network response. Furthermore, this altered response may be related to the clinical characteristics of the SRC athlete. [ABSTRACT FROM AUTHOR]

Fuzzy graph theory is applicable on diverse fields where uncertainty plays a significant role like in network analysis, medical diagnosis and health care, decision-making, and optimization, etc. An effective tool in this field is the topological index. These indices provide quantitative measures of molecular structure and connectivity, which are crucial for understanding and predicting physical, chemical, and biological properties of molecules. The Gutman index (GI) is an important topological index. This index is based on degree and distance. In this paper, GI is explored in fuzzy graphs. Some results on GI are found like the relationship of GI between a fuzzy graph and its fuzzy sub-graph and the relationship of GI between a fuzzy graph and its complete fuzzy graph. Some upper and lower bounds are obtained for complete fuzzy graph, bipartite graph, star, double star, etc. A relationship of GI between two isomorphic fuzzy graphs are determined. Human trafficking, a heinous offense and grave breach of human rights involves the coercion, deception, or coercion of people for a range of purposes, including forced work, forced marriage, and forced servitude. Overall, human trafficking is a complex and multifaceted issue that requires coordinated efforts at the local, national, and international levels to prevent, prosecute, and ultimately eradicate this egregious violation of human dignity. At the end of this paper, an application of GI is given to reduce human trafficking in India. [ABSTRACT FROM AUTHOR]

We can define six classes of endomorphisms on a graph, and they always form a chain based on set inclusion. The concepts of endomorphism type and endomorphism spectrum were introduced by Böttcher and Knauer in 1992. They provided a systematic and organized approach to study endomorphisms of graphs. In this paper, we focus on the double vertex wheel graphs and provide a detailed description of their endomorphism spectra and endomorphism types. [ABSTRACT FROM AUTHOR]

In this study, we introduce a novel index called the modified inverse sum indeg (MISI) energy as an extension of the traditional inverse sum indeg (ISI) energy and neighbor degree sum energy. We devised an algorithm that employs the simplified molecular input line entry system (SMILES) formula of compounds to compute the MISI energy of polycyclic aromatic hydrocarbon (PAH). Additionally, we established the bounds of the MISI energy for certain classes of graphs with fixed number of vertices. A strong correlation between the MISI energy and the total π-electron energy of PAHs is observed through regression analysis. Remarkably, this correlation outperforms that achieved by the classical ISI energy. These findings highlight the enhanced effectiveness of the MISI energy as a descriptor for capturing significant molecular properties. Moreover, our study establishes a foundation for further theoretical extensions and practical applications in the field of chemical graph theory. [ABSTRACT FROM AUTHOR]