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The document is a special issue of the Mathematics journal focusing on "Optimization Algorithms: Theory and Applications." It includes 10 selected papers from 9 countries covering topics such as arithmetic optimization, semi-parametric regression, flow shop scheduling, unconstrained optimization, metaheuristic techniques, many-objective optimization, and more. The papers present innovative algorithms and approaches that outperform existing methods in various optimization problems, showcasing the diverse research fields within the optimization area. [Extracted from the article]

This paper investigates the application of graph theory and variants of greedy graph coloring algorithms for the optimization of distributed peer-to-peer networks, with a special focus on private blockchain networks. The graph coloring problem, as an NP-hard problem, presents a challenge in determining the minimum number of colors needed to efficiently allocate resources within the network. The paper deals with the influence of different graph density, i.e., the number of links, on the efficiency of greedy algorithms such as DSATUR, Descending, and Ascending. Experimental results show that increasing the number of links in the network contributes to a more uniform distribution of colors and increases the resistance of the network, whereby the DSATUR algorithm achieves the most uniform color saturation. The optimal configuration for a 100-node network has been identified at around 2000 to 2500 links, which achieves stability without excessive redundancy. These results are applied in the context of a private blockchain network that uses optimal connectivity to achieve high resilience and efficient resource allocation. The research findings suggest that adapting network configuration using greedy algorithms can contribute to the optimization of distributed systems, making them more stable and resilient to loads. [ABSTRACT FROM AUTHOR]

Graph theory, a branch of mathematics, studies relationships among entities through vertices and edges. To capture the inherent uncertainties in real-world networks, Uncertain Graph Theory has evolved within this field. Soft Expert Graphs combine conventional graph theory with expert assessments, using fuzzy sets for vertices and edges, while allowing expert opinions to shape the uncertainties and relationships within the graph. Hypersoft Graphs extend this concept further by incorporating multi-attribute nodes that represent multiple distinct attribute values, enabling the modeling of more complex, multi-dimensional relationships. In this paper, we define the Hypersoft Expert Graph and explore its connections to other classes of graphs. We also consider the SuperHypersoft Graph, TreeSoft Graph, and IndetermSoft Graph. [ABSTRACT FROM AUTHOR]

The container terminal is a key node in global trade and logistics, where trucks connect quay cranes, storage yards, and vessels. Optimizing truck scheduling is crucial for enhancing port efficiency by addressing issues such as low truck utilization, excessive quay crane waiting times, and extended equipment completion times. This paper develops a container terminal simulation model based on graph theory, with the objective of minimizing the maximum completion time of terminal equipment. A collaborative scheduling algorithm for truck fleets, based on Deep Double Q-Networks (DDQN), is proposed. The algorithm designs five heuristic rules as the action space and refines state features and reward functions to optimize scheduling effectively. Experimental results indicate that this algorithm consistently identifies optimal scheduling strategies, outperforming both the five heuristic rules and the Deep Q-Network (DQN) algorithm. It significantly reduces quay crane waiting times and equipment completion times, improves truck utilization, and enhances overall container terminal efficiency. [ABSTRACT FROM AUTHOR]

This paper presents a resilience-based approach for critical load restoration in distribution networks using microgrids during extreme events when the main supply is disrupted. Reconfiguration of the distribution network using graph theory is investigated, for which Dijkstra's algorithm is first used to determine the shortest paths between microgrids and critical loads, and then the feasible restoration trees are established by combining the restorable paths. A mixed-integer linear programming (MILP) model is then used to find the optimal selection of feasible restoration trees to make a restoration scheme. The service restoration is implemented with the objectives of maximizing the energy delivered to the critical loads and minimizing the number of switching operations. The limited fuel storage of the generation sources in microgrids, the operational constraints of the network and microgrids, as well as the radiality constraint of the restored sub-networks, are considered the constraints of the optimization problem. The presented method can be used for optimal restoration of critical loads including the number of switching operations which is essential for the ease of implementation of a restoration plan. The results of simulations on a 118-bus distribution network demonstrate the efficiency of the procedure. [ABSTRACT FROM AUTHOR]

The powder bed fusion (PBF) process is increasingly employed by industry to fabricate complex parts with stringent standard criteria. However, fabricating parts "free of defects" using this process is still a major challenge. As reported in the literature, thermally induced abnormalities form the majority of generated defects, and are mainly the result of thermal evolution. Monitoring & controlling the temperature and its distribution throughout a layer under fabrication is an effective and efficient proxy to controlling such an evolution. In this paper, we introduce a novel online thermography and closed-loop hybrid-control (NOTCH)©, a practical control approach, to modify the scan strategy in metal PBF real-time. This system employs different mathematical thermophysical -based models, designed to optimize the printing sequence of different zones throughout a printing layer as well as the islands or stripes within each zone. Moreover, NOTCH uses artificial neural network (ANN) to optimize the energy density applied on each zone in order to avoid or mitigate some prevalent thermal anomalies. NOTCH strategy has two aims. First, producing a uniform temperature distribution throughout an entire layer to mitigate the thermally induced residual stress and its related distortion. In this step, we optimize the printing sequence of islands or stripes in their respective scan strategies. This paper expands on three potential models, explains pros and cons of these models, and presents preliminary results for a printed prototype. Second, controlling the laser specifications in order to avoid heat affected zones (HAZ) and mitigate thermal abnormalities such as the balling phenomenon. This step enables a smart adjustment of energy density by using ANN to avoid or mitigate HAZs, generate uniform microstructure with minimum porosity, and contribute to a more uniform temperature distribution. The completion of the latter step is part of the ongoing research which should be reported in future publications. [ABSTRACT FROM AUTHOR]

In this paper, we study perfect 2-coloring of the six regular graphs up to order 10. We first obtain all possible color parameter matrices and then examine them according to the limitation of our theorem. [ABSTRACT FROM AUTHOR]

This paper investigates the properties of tree-width and related graph width parameters for n-SuperHyperGraphs, a broader generalization of hypergraphs. By exploring concepts such as SuperHyperTreewidth and Hypertree-width, we aim to understand how these structures resemble tree-like formations. We also analyze the relevance of these width parameters in computational contexts, following extensive research in graph theory and hypergraph applications. [ABSTRACT FROM AUTHOR]

Graph theory is a fundamental area that explores the properties and relationships present within graph structures. This paper explores generalized graph complements by examining edge partitioning, a novel approach that extends the traditional graph complement. In analogy to the vertex partition-based definitions of generalized complements, two distinct types of generalized complements for a graph emerge through edge partitioning: k'-complement and k'(i)- complement of a graph. k'-complement of a graph emerges as a versatile tool with applications across multiple domains, due to its maximum edge configuration linked with a Fibonacci polynomial and its role as a maximal outerplanar graph. The paper systematically examines theorems and fundamental properties governing the structural relationships between the original graph and its generalized complements. The characterizations of cycles and paths are also analysed in this study. [ABSTRACT FROM AUTHOR]

Domination problems are fundamental problems in graph theory with diverse applications in optimization, network design, and computational complexity. This paper investigates { k } -domination, k-tuple domination, and their total domination variants in weighted strongly chordal graphs and chordal bipartite graphs. Specifically, the { k } -domination problem in weighted strongly chordal graphs and the total { k } -domination problem in weighted chordal bipartite graphs are shown to be solvable in O (n + m) time. For weighted proper interval graphs and convex bipartite graphs, we solve the k-tuple domination and total k-tuple domination problems in O (n 2.371552 log 2 (n) log (n / δ)) , where δ is the desired accuracy. Furthermore, for weighted unit interval graphs, the k-tuple domination problem achieves a significant complexity improvement, reduced from O (n k + 2) to O (n 2.371552 log 2 (n) log (n / δ)) . These results are achieved through a combination of linear and integer programming techniques, complemented by totally balanced matrices, totally unimodular matrices, and graph-specific matrix representations such as neighborhood and closed neighborhood matrices. [ABSTRACT FROM AUTHOR]

Molodtsov pioneered the notion of soft set theory, presenting it as a mathematical tool for dealing with uncertainty. Numerous researchers have subsequently developed models leveraging this theory to tackle challenges in decision-making and medical diagnosis. Soft set theory emerges as a flexible framework adept at handling uncertain and imprecise information, a domain where classical set theory often struggles. Expanding on the soft set concept, researchers have introduced the idea of a soft graph. This innovative concept allows for the creation of diverse representations of graph-based relations by incorporating parameterisation. In this work, we present and investigate some of the features of the homomorphic and restricted homomorphic products of soft graphs. This paper establishes the structural properties of these products, ensuring that they are well-defined and maintain the essential characteristics of soft graphs. Additionally, we derive combinatorial identities related to the counts of vertices and edges, as well as the degree sums, offering deeper insights into the composition and behaviour of these graph products. [ABSTRACT FROM AUTHOR]

Pre-trained neural networks like GPT-4 and Llama2 have revolutionized intelligent information processing, but their deployment in industrial applications faces challenges, particularly in harsh environments. To address these related issues, model offloading, which involves distributing the computational load of pre-trained models across edge devices, has emerged as a promising solution. While this approach enables the utilization of more powerful models, it faces significant challenges in harsh environments, where reliability, connectivity, and resilience are critical. This paper introduces failure-resilient inference in mobile networks (FRIM), a framework that ensures robust offloading and inference without the need for model retraining or reconstruction. FRIM leverages graph theory to optimize partition redundancy and incorporates an adaptive failure detection mechanism for mobile inference with efficient fault tolerance. Experimental results on DNN models (AlexNet, ResNet, VGG-16) show that FRIM improves inference performance and resilience, enabling more reliable mobile applications in harsh operating environments. [ABSTRACT FROM AUTHOR]

This paper investigates Pendant Graphs, which are graphs that contain pendant vertices. It introduces a new perspective on the regularity and irregularity of pendant graphs, and discusses their relationship with Highly Irregular Graphs. Additionally, this work introduces a novel graph based on the concept derived from (m, k) Regular Graphs. The (m, k) regular graphs are closely related to Pendant Graph. The paper also delves into the nature and characteristics of pendant vertices. By exploring the characteristics of pendant vertices and defining the Cyclic Dendrimer Graph, the paper significantly contributes to both Graph Theory and Chemical Graph Theory. [ABSTRACT FROM AUTHOR]

At present, the method of using rule set partitioning technology to assist in constructing multiple decision trees for packet classification has been widely recognized. Rule set partitioning demonstrates a unique symmetry-breaking mechanism, systematically transforming the initial overlapping rule space into a more structured and balanced configuration. By separating overlapping rules in the initial stage, this method significantly reduces rule replication within trees, thereby improving the algorithm's classification performance. The asymmetric characteristics of this partitioning process are particularly noteworthy: through the strategic disruption of the initial rule set's symmetric distribution, it creates asymmetric subspaces with enhanced computational efficiency. However, existing research lacks standardized metrics for evaluating the effectiveness of rule set partitioning schemes. The purpose of this paper is to investigate the impact of partitioning on algorithm performance. Based on community structure theory, we construct a weighted graph model for rule sets and propose a disassortative modularity metric to evaluate the effectiveness of rule set partitioning. This metric not only examines intra-community connections but also emphasizes the asymmetric connections between communities. By quantifying these structural features, it provides a novel perspective on rule set partitioning strategies. The experimental results demonstrate a significant positive correlation between disassortative modularity and classification throughput. This metric offers valuable guidance for packet classification partitioning techniques, highlighting the practical significance of symmetry and asymmetry in algorithm design. [ABSTRACT FROM AUTHOR]

The impact of disturbances on a transportation network varies depending on the location and characteristics of the affected highway segments. Given limited resources, it is crucial to prioritize the protection and repair of highway segments based on their importance to maintaining overall network performance during disruptions. This paper proposes a novel method for ranking the importance of highway segments, leveraging a novel local–transit percolation and clustering-based method. Initially, the highway network is constructed by Graph theory, and the k-means clustering method is applied considering each segment's transit and local traffic flows. Subsequently, a local–transit percolation model is constructed to generate an initial ranking of segments based on the size of the second-largest clusters during the percolation phase transition. A secondary ranking is performed by refining the results from the clustering phase. Results of a control experiment show that, compared to baselines, the proposed ranking approach demonstrates a significantly improved ability to sustain network demand and connectivity when high-ranked segments are moved. The model uncertainty analysis was conducted by adding noise to the gantry records, and the experiments demonstrated that the model exhibits robustness under noisy conditions. These findings highlight the effectiveness and superiority of the proposed method. [ABSTRACT FROM AUTHOR]

In this paper, we explore the connection between sensor networks and graph theory. Sensor networks represent distributed systems of interconnected devices that collect and transmit data, while graph theory provides a robust framework for modeling and analyzing complex networks. Specifically, we focus on vertex coloring, Eulerian paths, and Hamiltonian paths within the Delaunay graph associated with a sensor network. These concepts have critical applications in sensor networks, including connectivity analysis, efficient data collection, route optimization, task scheduling, and resource management. We derive theoretical results related to the chromatic number and the existence of Eulerian and Hamiltonian trails in the graph linked to the sensor network. Additionally, we complement this theoretical study with the implementation of several algorithmic procedures. A case study involving the monitoring of a sugarcane field, coupled with a computational analysis, demonstrates the performance and practical applicability of these algorithms in real-world scenarios. [ABSTRACT FROM AUTHOR]

Let G be a finite simple graph and let A (G) be its adjacency matrix. Then, G is singular if A (G) is singular. The singularity of graphs is of certain interest in graph theory and algebraic combinatorics. For positive integers a i ≥ 2 , i = 1 , 2 , ... , 6 . Insert a 1 − 2 , a 2 − 2 , a 3 − 2 , a 4 − 2 , a 5 − 2 and a 6 − 2 vertices in the six edges of the complete graph K 4 , respectively, then the resulting graph is called the K 4 homeomorphic graph, denoted by K (a 1 , a 2 , a 3 , a 4 , a 5 , a 6) . In this paper, we give the necessary and sufficient condition for the singularity of K (a 1 , a 2 , a 3 , a 4 , a 5 , a 6) , and we also show that the probability of a K 4 homeomorphic graph K (a 1 , a 2 , a 3 , a 4 , a 5 , a 6) being a singular graph is equal to 193 512 . [ABSTRACT FROM AUTHOR]

A new area of study called "neutrosophic graph theory" uses neutrosophic logic to expand on traditional graph theory to include ambiguous, indeterminate, and uncertain information. This paper gives a thorough introduction to neutrosophic graphs, including their definition, components, uses, and difficulties. Neutrosophic graphs let you show and analyze relationships that aren't clear-cut because they give vertices and edges different levels of truth, indeterminacy, and falsity. Image processing, social network analysis, pattern recognition, and decision-making in the face of uncertainty all use neutrosophic graphs. However, we still need to resolve issues like interpretability and computational complexity. All things considered, neutrosophic graphs provide a viable framework for representing and evaluating ambiguous information in graph structures, with potential applications in a variety of fields. [ABSTRACT FROM AUTHOR]

The line graph is a very popular research object in graph theory, in complex networks and also in social networks recently. Let L (K m , n) be the line graph of the complete bipartite graph K m , n and P k + 1 be a path of length k. In this paper, we give necessary and sufficient conditions for the existence of P 4 -decompositions of L (K m , n). [ABSTRACT FROM AUTHOR]

This paper explores Iwasawa theory from a graph-theoretic perspective, focusing on the algebraic and combinatorial properties of Cayley graphs. Using representation theory, we analyze Iwasawa-theoretic invariants within Z ℓ -towers of Cayley graphs, revealing connections between graph theory, number theory, and group theory. Key results include the factorization of associated Iwasawa polynomials and the decomposition of μ - and λ -invariants. Additionally, we apply these insights to complete graphs, establishing conditions under which these invariants vanish. [ABSTRACT FROM AUTHOR]

The study of spectral graph determination is a fascinating area of research in spectral graph theory and algebraic combinatorics. This field focuses on examining the spectral characterization of various classes of graphs, developing methods to construct or distinguish cospectral nonisomorphic graphs, and analyzing the conditions under which a graph's spectrum uniquely determines its structure. This paper presents an overview of both classical and recent advancements in these topics, along with newly obtained proofs of some existing results, which offer additional insights. [ABSTRACT FROM AUTHOR]

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The problem of finding the largest cluster in an undirected simple bipartite graph with the maximum number of edges is known as Maximum Edge Bi-clique (MEB) problem. MEB has many applications in several fields like bioinformatics, social networks and data mining. Since the problem has been proved to be NP-complete, the Binary Linear Programming (BLP) is an interesting alternative for solving it. In this paper, a new BLP model is proposed which has fewer variables than previous models which finds a bi-clique with the given size, if exist. First, by identifying and removing redundant constraints from previous models, the BLP model becomes more compact, which in some cases allows it to find larger bi-cliques, improving the overall solution quality. Then, a more efficient quaternary search method is proposed to find the MEB. Afterward, a theorem is proved which helps to reduce the search space of the quaternary search and as a result to reduce the number of calls to the BLP model. Then another theorem is proved which helps to propose 3 models which we call it 3-Binary Linear Programming (3-BLP) procedure. 3-BLP is especially efficient for bipartite graphs in which the difference between the maximum degrees of nodes in two sides of the bipartite graph is large. The experimental results show the superiority of the proposed approaches in terms of speed and accuracy compared to the state of the art. [ABSTRACT FROM AUTHOR]

Since its inception, the femtosatellite has developed into a complex and capable spacecraft. However, despite such advances the implementation of predicted applications remains elusive. In response, this paper explores three femtosatellite mission architectures, the drone, swarm, and hive, to characterize what possible femtosatellite mission could be envisioned for true space applications. Starting from a single femtosatellite, each architecture represents an increase in network complexity, so the use of graph theory is proposed as a means for characterizing, understanding, and modelling femtosatellite systems. A simple reliability model is also proposed for each of the mission architectures, and a simulation is carried out to generate a profile of the system's lifecycle. The results are discussed for each of the architectures in relation to potential applications. Each of the architectures was shown to be feasible for short duration missions due to the inherent unreliability of femtosatellites. Additionally, mid- and long-duration missions were shown to be possible for swarms and hives with sufficiently large populations. Finally, applications are re-examined considering the lifecycle profile exhibited by each system. • Femtosatellite mission architectures are introduced, each representing an increased level of complexity and capability. • A mission assurance strategy is established for the characterization of femtosatellite mission lifecycles. • A mission lifecycle is predicted from simulation and reliability calculations, and application recommendations are made. [ABSTRACT FROM AUTHOR]

The preface to the special issue "Algebraic Structures and Graph Theory, 2nd Edition" highlights the publication of ten articles and one review paper in the Mathematics journal, showcasing original research advancements in algebraic structures and graph theory. The selected papers cover topics such as graphs from lattices, semigroups, and groups, edge-primitive graphs, friendship graphs, equitable graphs, and hypercompositional algebras. Contributions include characterizations of various graph types, planarity discussions, and structure theorems for different algebraic structures, providing valuable insights into these mathematical areas. The authors express gratitude to contributors and reviewers for their valuable input, enhancing the quality of the published manuscripts. [Extracted from the article]

Drones have recently become more popular due to technological improvements that have made them useful in many other industries, including agriculture, emergency services, and military operations. Coordination of communication amongst drones is often required for the efficient performance of missions. With an emphasis on building robust k -connected networks and restoration procedures, this paper investigates the relevance of connection in drone swarms. Specifically, we tackle the k -connectivity restoration problem, which aims to create k -connected networks by moving the drones as little as possible. We propose four novel approaches, including an integer programming model, an integer programming-based heuristic approach, a node converging heuristic, and a cluster moving heuristic. Through extensive measurements taken from various drone networking setups, we provide a comparative analysis of the proposed approaches. Our evaluations reveal that the drone movements produced by the integer programming-based heuristics are nearly the same as the original mathematical formulation, whereas the other heuristics are favorable in terms of execution time. • We study on the k-connectivity restoration problem for drone networks. • We propose 4 novel approaches to tackle the target problem. • Proposed approaches are evaluated on extensive drone networking setups. • The results of proposed integer programming-based heuristic are nearly same as the original mathematical formulation. • Other proposed heuristics are favorable in terms of execution time. [ABSTRACT FROM AUTHOR]

This paper presents a comprehensive framework for spatiotemporal flow field analysis based on complex network theory, emphasizing dimensionality reduction, spatiotemporal feature identification, modeling, and sparsification. The framework first redefines transient flow fields using graph theory and applies clustering techniques to discretize the flow field into different vortex structures, achieving dimensionality reduction. By leveraging the structural relationships between nodes and edges in the network, this method establishes the physical correlations between vortices. It effectively describes flow characteristics by transforming the time evolution of physical properties into the motion of network nodes, modeling both the static and dynamic aspects of the flow field. A community detection approach based on dynamically linked node attributes reveals different flow states and their evolution patterns. Moreover, the model, implemented through network dynamics, successfully predicts unsteady forces and provides complete flow field information. The use of superpixel-based sparse representation of the flow field strikes a balance between simplicity and the preservation of key flow features. This approach highlights the growing importance of sparsity in vortex and flow state identification while also emphasizing the delicate balance between data simplification and accuracy during modeling. This innovative method significantly simplifies flow field analysis and offers a comprehensive insight into its dynamic behavior, providing a promising framework for understanding and characterizing complex transient flow phenomena. [ABSTRACT FROM AUTHOR]

In this paper, we present an alternative method to grain growth simulations. Traditional grain growth algorithms can be computationally expensive, especially when considering anisotropic grain boundary (GB) properties. The new Semi-Stochastic Grain Growth Prediction (SSGGP) model consists of two main components: a statistical evolution model that predicts the evolution of the GB network spectrum and a conditional diffusion model that generates grain growth morphologies at different time steps. These models are trained on a dataset Niño and Johnson (2024) that contains thousands of microstructures obtained from anisotropic grain growth simulations. We test the effectiveness of our model by comparing microstructure statistics (e.g., grain size distribution, orientation distribution function (ODF), misorientation distribution function (MDF), and GB energy distribution) with those obtained from grain growth simulations. The results indicate that the SSGGP model shows good agreement in terms of these statistics. Moreover, once trained, the SSGGP is almost ten times faster in obtaining the evolved state of a microstructure. We also find evidence for self-similarity of the GB network during steady-state normal anisotropic grain growth. [Display omitted] [ABSTRACT FROM AUTHOR]

This article is concerned with the study of the existence of a distributional solution for a strongly nonlinear (p(x), q(x))--elliptic systems. By means of the Berkovits degree theory, with suitable assumptions on the nonlinearities, we prove the existence of nontrivial solutions to our problem. [ABSTRACT FROM AUTHOR]

In graph theory, the hypergraph [22] extends the traditional graph structure by allowing edges to connect multiple vertices, and this concept is further broadened by the superhypergraph [174,176]. Additionally, several types of uncertain graphs have been explored, including fuzzy graphs [136, 153], neutrosophic graphs [35, 36], and plithogenic graphs [66, 75, 185]. This study explores the SuperHyperGraph, Single-Valued Neutrosophic Quasi SuperHyperGraph, and Plithogenic Quasi SuperHyperGraph, analyzing their relationships with other graph classes. Future work will define the Semi Superhypergraph, Multi Superhypergraph, Pseudo Superhypergraph, Mixed Superhypergraph, and Bidirected Superhypergraph and examine their connections to existing classes in hypergraphs and graphs. [ABSTRACT FROM AUTHOR]

This study aimed to evaluate the effects of a 12-week intervention combining electroacupuncture (EA) with conventional cognitive rehabilitation (CR) on cognitive recovery and brain network topological properties in stroke patients. Thirty-four post-stroke cognitive impairment (PSCI) patients were randomly assigned to either an EA + CR group or a CR-only group, with both undergoing five weekly interventions for 12 weeks. Cognitive assessments and Diffusion Tensor Imaging tests were performed pre- and post-intervention. Primary outcomes included the Montreal Cognitive Assessment (MoCA), Digit Span Test (DST), Auditory Verbal Learning Test (AVLT-H), and Aphasia Screening Scale. Secondary outcomes measured brain network global and nodal properties. The EA + CR group showed significant improvements in MoCA (p < 0.001), AVLT-H (p = 0.004), and in areas like naming (p = 0.022), attention (p < 0.001), and delayed recall (p < 0.001). Local brain network metrics, such as clustering coefficients (Cp) and local efficiency (Eloc), were significantly higher (p < 0.05) in the EA + CR group, though global efficiency (Eglob) was unchanged. Improvements in Eloc were positively correlated with MoCA scores (r = 0.0716, p = 0.018). Specific brain regions, including the medial and paracentral cingulate gyrus, inferior temporal gyrus, and left supramarginal gyrus, exhibited significant nodal property differences, suggesting that EA may enhance cognitive recovery by targeting cognitively relevant regions. These findings indicate that EA, combined with CR, is a safe and effective treatment for cognitive impairment following stroke, potentially via structural brain network improvements. Clinical trial registration: https://clinicaltrials.gov/ , ChiCTR2200066160. [ABSTRACT FROM AUTHOR]

During the pandemic, decision-makers offered many preventive policies to reduce the negative effects of the pandemic. The social distance rule in enclosed areas was implemented by educational institutions in any countries. In this study, we deal with the problem of assigning students to seats by considering the social distancing constraint and with objective of maximizing the total distance among the students. This problem is found to be similar to the Maximum Diversity Problem (MDP) in the literature. We name this new problem as Maximum Diversity Social Distancing problem (MDPs). A simulated annealing algorithm framework for MDPs (SA-MDPs) is proposed to identify an optimal or near-optimal solution within a reasonable computational time. A greedy random-based algorithm is presented to determine efficiently an initial feasible solution. The new neighborhood search procedure based on graph theory is introduced, in which the dominated, dominating, and nondominated seats are determined based on social distance. The proposed SA-MDPs is evaluated on classrooms with varying capacities and benchmarked against an off-the-shelf optimization solver. The computational tests demonstrated that the SA-MDP model consistently provided either proven optimal solutions or superior best-known solutions compared to a commercial solver, all within a reasonable CPU time. [ABSTRACT FROM AUTHOR]

In this manuscript, we present several novel results in fixed-point theory for a complete controlled metric space. The first presented result is inspired from the Caristi contraction where we explore the existence and uniqueness of fixed points under specific conditions. Furthermore, we propose a graphical form of it by endowing the considered space with a graph and develop a new fixed-point theorem, which is illustrated by two examples. Also, we establish a theorem for the α -admissible mapping. To demonstrate its effectiveness, the last theorem proposes an approach to solve a second-order differential equation. [ABSTRACT FROM AUTHOR]

An n-vertex graph is said to to be (p,β)-bijumbled if for any vertex sets A,B⊆V(G), we have e (A,B) = p∣A∣∣B∣±β√∣A∣∣B∣. We prove that for any r∈N ≥3 ​ and c>0 there exists an ε>0 such that any n-vertex (p, β)-bijumbled graph with n∈rN, p> 0, δ(G) ≥ cpn and β≤εpr−1n contains a Kr -factor. This implies a corresponding result for the stronger pseudorandom notion of (n,d,λ)-graphs. For the case of triangle factors, that is, when r=3, this result resolves a conjecture of Krivelevich, Sudakov and Szabó from 2004 and it is tight due to a pseudorandom triangle-free construction of Alon. In fact, in this case even more is true: as a corollary to this result and a result of Han, Kohayakawa, Person and the author, we can conclude that the same condition of β=o (p²n) actually guarantees that a (p,β)-bijumbled graph G contains every graph on n vertices with maximum degree at most 2. [ABSTRACT FROM AUTHOR]

Introduction: Neuroimaging studies have demonstrated that intranasal oxytocin has extensive effects on the resting state functional connectivity of social and emotional processing networks and may have therapeutic potential. However, the extent to which intranasal oxytocin modulates functional connectivity network topology remains less explored, with inconsistent findings in the existing literature. To address this gap, we conducted an exploratory data-driven study. Methods: We recruited 142 healthy males and administered 24 IU of intranasal oxytocin or placebo in a randomized controlled double-blind design. Resting-state functional MRI data were acquired for each subject. Network-based statistical analysis and graph theoretical approaches were employed to evaluate oxytocin's effects on whole-brain functional connectivity and graph topological measures. Results: Our results revealed that oxytocin altered connectivity patterns within brain networks involved in sensory and motor processing, attention, memory, emotion and reward functions as well as social cognition, including the default mode, limbic, frontoparietal, cerebellar, and visual networks. Furthermore, oxytocin increased local efficiency, clustering coefficients, and small-world propensity in specific brain regions including the cerebellum, left thalamus, posterior cingulate cortex, right orbitofrontal cortex, right superior frontal gyrus, left inferior frontal gyrus, and right middle orbitofrontal cortex, while decreasing nodal path topological measures in the left and right caudate. Discussion: These findings suggest that intranasal oxytocin may produce its functional effects through influencing the integration and segregation of information flow within small-world brain networks, particularly in regions closely associated with social cognition and motivation. [ABSTRACT FROM AUTHOR]

Background: Over the past decades, the analysis metabolic connectivity patterns has received significant attention in exploring the underlying mechanism of human behaviors, and the neural underpinnings of brain neurological disorders. Brain network can be considered a powerful tool and play an important role in the analysis and understanding of brain metabolic patterns. With the advantages and emergence of metabolic-based network analysis, this study aims to systematically review how brain properties, under various conditions, can be studied using Fluorodeoxyglucose Positron Emission Tomography (FDG-PET) images and graph theory, as well as applications of this approach. Additionally, this study provides a brief summary of graph metrics and their uses in studying and diagnosing different types of brain disorders using FDG-PET images. Main body: In this study, we used several databases in Web of Science including Web of Science Core Collection, MEDLINE to search for related studies from 1980 up to the present, focusing on FDG-PET images and graph theory. From 68 articles that matched our keywords, we selected 28 for a full review in order to find out the most recent findings and trends. Our results reveal that graph theory and its applications in analyzing metabolic connectivity patterns have attracted the attention of researchers since 2015. While most of the studies are focusing on group-level based analysis, there is a growing trend in individual-based network analysis. Although metabolic connectivity can be applied to both neurological and psychiatric disorders, the majority of studies concentrate on neurological disorders, particularly Alzheimer's Disease and Parkinson's Disease. Most of the findings focus on changes in brain network topology, including brain segregation and integration. Conclusion: This review provides an insight into how graph theory can be used to study metabolic connectivity patterns under various conditions including neurological and psychiatric disorders. [ABSTRACT FROM AUTHOR]

The category of directed graphs is isomorphic to a particular category whose objects are labeled undirected bipartite graphs and whose morphisms are undirected graph morphisms that respect the labeling. Based on this isomorphism, we begin by showing that the class of all directed graphs is a Graph Isomorphism Complete class. Afterwards, by extending this categorical framework to weighted prime graphs, we prove that the categories of multidirected graphs with and without self-loops are each isomorphic to a particular category of weighted prime graphs. Consequently, we prove that these classes of multidirected graphs are also Graph Isomorphism Complete. [ABSTRACT FROM AUTHOR]

In this article, we present the concept of extended Seidel energy by employing a generalized extended matrix to study various molecular properties, including the Kovats retention index, boiling point, enthalpy of formation, entropy, acentric factor, and octanol-water partition coefficient. Our research broadens the scope of energy matrices in graph theory, with a particular emphasis on Sombor energy, reduced Sombor energy, average Sombor energy, Banhatti Sombor energy and reduced Banhatti Sombor energy. We examined the correlation of these graph-based energy descriptors with the thermodynamic properties of Benzenoid hydrocarbons (BHC), uncovering strong relationships between these indices and different molecular attributes. These results provide valuable insights into the predictive potential of graph energies in modeling the thermodynamic behavior of polycyclic aromatic hydrocarbons. [ABSTRACT FROM AUTHOR]

An algebraic graph is defined in terms of graph theory as a graph with related algebraic structures or characteristics. If the vertex set of a graph G is a group, a ring, or a field, then G is called an algebraic structure graph. This work uses an algebraic structure graph based on the modular ring Z n , known as a hyper-chordal ring network. The lower and upper bounds of the local fractional metric dimension are computed for certain families of hyper-chordal ring networks. Utilizing the cardinalities of local fractional resolving sets, local fractional resolving (LFR)M-polynomials are computed for hyper-chordal ring networks. Further, new topological indices based on (LFR)M-polynomials are established for the proposed networks. The local fraction entropies are developed by modifying the first three kinds of Zagreb entropies, which are calculated for the chosen hyper-chordal ring networks. Furthermore, numerical and graphical comparisons are discussed to observe the order between newly computed topological indices. [ABSTRACT FROM AUTHOR]