Graph Theory and Molecular Behavior: Utilizing Platt Numbers and Polynomial Approximations for Boiling Point Estimations

The use of graph‐theoretic parameters is essential in interpreting the structural robustness, efficiency, and functionality of complex networks. Specifically, edge‐based metrics are more useful in interpreting the susceptibility of links and the concept of upstream impact, as well as structural robustness. Such metrics are essential in a variety of fields such as network security, communication, and optimization theory. In the same way, a new edge‐based invariant is introduced in this research paper, which is known as the b -edge differential . The b‐edge differential is essentially linked to b ‐differential of a graph. Let G = ( V , E ) be a simple connected graph. For any subset of edges X ⊆ E , we associate the set B ( X ) = { e ∈ E \ X : e shares a common vertex with at least one edge in X }, that is, B ( X ) captures all edges outside X which are adjacent to X . The quantity ∂ B E ( X ) = | B ( X )| is called the b ‐edge differential of X . We then define the b ‐edge differential of a graph G by ∂ B E ( G ) = max X ⊆ E ∂ B E ( X ). This parameter calculates the maximum value of the indirect dominances, or coverage, which can be achieved by a given set of edges. It serves as a useful tool in the identification of critical edges as well as the maximization of influence in networks. The problem can also be used in the applications involving the maximization of profit, traffic routing, and vulnerability evaluation for communication networks. We carry out the explicit computation of the b ‐edge differential for some classical types of graphs, such as paths, cyclic graphs, wheel graphs, complete graphs, star graphs, double‐star graphs, comb graphs, complete bipartite graphs, ladder graphs, and triangular ladder graphs. The work described in this article expands the relevance of differential parameters in the theory of graphs, with promising research directions for theoretical and application aspects of network science.

A vascular network is often represented by a Reeb graph, which is a topological skeleton, and graph theory has been widely applied to analyze properties of a vascular network. A Reeb graph model for a vascular network is obtained by assigning the branch points of the network to be the vertices of the graph and the vessels between branch points to be the edges of the graph. Vascular networks develop by way of angiogenesis, a growth process that involves the biological mechanisms of vessel sprouting (budding) and splitting (intussusception). Vascular networks develop by way of two biological mechanisms of vessel sprouting (budding) and splitting (intussusception). According to a graph theory modeling of two vascular network growth mechanisms, all nodes in the Reeb graph must be cubic in degree except for two special nodes: the afferent (A) and efferent (E) nodes. We define that a vascular network is cubic if all internal nodes are cubic in degree. We consider six normal adult rat renal glomerular networks and use their reeb graphs already constructed and published in the literature. We observe that five of them contain internal vertices of degree higher than three. Branch points in vascular networks may appear to be of a higher degree if the imaging resolution cannot differentiate between blood vessels that are very close in proximity. Here, we propose a random graph theory model that edits a non-cubic vascular network into a cubic graph. We observe that the edited cubic graph from a non-cubic vascular network has the similar size and order as the one cubic vascular network.

IntroductionA variety of models, methods, and computer-aided systems have been used to predict and analyze disasters and emergencies, and manage a pertinent medical response effectively. The list of these instruments traditionally comprises: computerized databases, geographic information systems (GISs), graph theory, complex networks, mathematical programming, simulation, and agent-based modeling. As a rule, each type of those is applied asunder. It makes sense to put in one silo diverse instruments to get multidisciplinary solution with its synergy effect for problems of disaster and emergency medicine.MethodsWithin this study, an original, agent-based model was developed. The model combines the advantages of the principal computer-aided instruments and considers all the types of information: semantic, topographical, metric, and topological. The model is severely dynamic, fits to real actors and principally covers all the disaster situation.ResultsIn line with the model, a so called Topometric Agent-Based System (TABS) with its key visualization component has been designed. A TABS-specific simulation to investigate behavior of the attacked network of vulnerable actors has been conducted leading to critical findings. It has been shown a severe significance of order within combination of threats: man-made + natural or those of natural + man-made. A well-balanced financial distribution to protect actors of diverse status also has been found.ConclusionsA TABS similar to GIS focuses on mapping. Topometric Agent-Based System mapping brings an efficient and clear language for information sharing not only within national emergency medical services but between experts from different fields and countries.

BACKGROUND Radiation necrosis (RN) is a debilitating, delayed complication of radiotherapy characterized by progressive neurological decline and limited treatments. Despite its clinical significance, the molecular mechanisms underlying RN remain poorly defined, hampering the development of targeted therapies. This study employed an integrative multi-omics approach to uncover conserved molecular signatures and identify potential therapeutic targets in RN. METHODS RN tissue samples were collected from patients with diverse primary brain lesions (gliomas, metastases, AVMs). Histologically confirmed RN and matched non-necrotic tissues underwent RNA sequencing (RNA-Seq) and mass spectrometry-based proteomics (MS). Graph theory and partial least squares (PLS) modeling were used for network characterization and multi-omics integration. Complementary in-vivo studies involved whole-brain irradiation of mice, followed by a latency period and cognitive assessment using Morris water maze. Mouse brains were then analyzed (RNA-Seq & MS). RESULTS Unsupervised clustering revealed a distinct molecular profile in RN tissue, independent of primary lesion. A total of 678 differentially expressed genes (DEGs) and 153 differentially abundant proteins (DAPs) were identified. RN tissue showed upregulation of pro-inflammatory genes, complement activation, extracellular matrix remodeling, and hypoxia-associated pathways. Integrated analysis identified 14 concordantly upregulated DEG and DAP, (including C3, APOD, IGHG2), as well as 6 DEG/DAP pairs with discordant expression patterns. Network analysis revealed non-random hub connectivity (including STAT3, FN1, SPP1, TIMP1)—genes implicated in inflammation and tissue remodeling. PLS-based multi-omics integration identified a robust RN-specific molecular signature that strongly correlated with radiographic cerebral edema (R²=0.8,P<0.001). Survival analysis demonstrated that patients with higher dysregulation of network-core DEGs/DAPs had a 5.1-fold increased mortality risk(P<0.001). In-vivo, irradiated mice exhibited significant impairments in spatial memory tasks(P<0.05), and proteomic profiling revealed marked changes in protein expression. CONCLUSION This comprehensive multi-omics analysis defines the molecular architecture of RN, identifies central regulatory nodes, and provides a foundation for biomarker-guided therapeutic development.

In this study, we present a scenario-based approach to investigate the protest potential in society using mathematical tools from graph theory and simulation modeling. The research aims to construct a cognitive map that reflects key factors, such as population deprivation levels, overall well-being, social fears, trust in authority, social expectations, and stability of social structures. We report the results of a scenario analysis of the developed model, obtaining forecasted scenarios of the dynamics of society’s protest potential under various conditions and influences. At the core of this research is the examination of a complex system’s possible response to external destructive impacts, and we propose a set of measures to counteract these influences on societal dynamics. The scenario methodology combines graph theory for representing complex relationships between key factors and simulation modeling for predicting potential developmental scenarios. Conducting this study required modifying the traditional mathematical apparatus of cognitive modeling. We developed methods and algorithms for scenario-event identification of the model’s significant factors’ behavior and utilizing the results of such identification when forming complex functional relationships in the scenario model. In generating scenarios, the range of events is significantly expanded compared to traditional cognitive modeling approaches, which could have been overlooked but may be crucial for situational analysis.

Indices of trophic and competitive relations in a planktonic network of a shallow, temperate lagoon. A graph and structural equation modeling approach

The modeling of a modular robot system is complex due to its capability of changing and forming countless robot configurations with a set of modules. The greater the number of modules added to the structure, more complicated is the robot and its modeling. A methodology to represent modular robot configurations as a fixed shape robot is presented. The equivalent model is obtained through the characterization of the module, graph theory and geometric modeling of robots. After the transformation, the resulting kinematic and dynamic parameters are similar to the traditional model of a single configuration robot. Once the equivalent model is obtained, it can be used within different robot applications.

Let be a graph with p vertices and q edges. We have introduced a new graph labeling method using integers and cordial-related works and investigated some graphs for this labeling technique. Using this labeling concept, we have examined the graphs like path, cycle, star, complete graph, comb, and wheel graph. The first research paper on graph theory was published by Leonhard Euler. However, he did not use the word ‘graph’ in his work. In the early stages of the development of the subject, the vertices of a graph were specified as , and the edges were denoted by, . In recent times, several researchers have attempted to provide different types of labeling to the vertices and edges of a graph by identifying the relevant mathematical properties. The present paper provides a novel method of labeling by employing integers, which may form a foundation for future research work. In this paper, we investigate the pair mean cordial labeling behavior of some cycle-related graphs like the ice cream graph, closed web graph, circulant graph, zig-zag chord graph, pentagonal circular ladder, djembe graph, quadrilateral friendship graph, and origami graph.

This paper presents first an analysis of the topology of mechanisms via Graph Theory and Screw Theory and next the principle of dual mechanisms in terms of their mobility and overconstraints. Using dual graphs, the graph representations of the mechanisms that are dual to hybrid and Delta mechanisms are revealed. The concept of parallel mechanisms with configurable platforms (PMCPs) is introduced and it is shown that the graph reduction of PMCPs always results in a wheel graph, which has the interesting property of being self-dual. In case of self dual topology, it is then possible to directly convert any method developed for their mobilty analysis into an overconstraint analysis method and vice versa. This self dual topology property can also be exploited to create new PMCPs and is an important aspect in the future development of a type synthesis method that includes PMCPs.KeywordsParallel mechanismsConfigurable platformTopologyMobilityOverconstraintsDuality

This dissertation solves two problems relating to the structure of graphs. The first of these is motivated by Kuratowski's Theorem, perhaps the most famous result in graph theory. This theorem states the $K\sb5$ and K$\sb{3,3}$ are the only non-planar graphs for which both $G\\ e$, the deletion of the edge e, and G/e, the contraction of the edge e, are planar for all edges e of G. We characterize the non-planar graphs for which $G\\ e$ or G/e is planar for all edges e of G. The second problem we solve is motivated by Tutte's wheels theorem. An immediate consequence of this theorem is that the wheel graphs are the basic building blocks for the collection of simple, 3-connected graphs. Therefore it is of interest to examine the structure of the graphs that do not have a minor isomorphic to the k-spoked wheel, $W\sb{k}$, for small values of k. Dirac determined that the graphs having no $W\sb3$-minor are the series-parallel networks. It follows easily from Tutte's wheels theorem is that $W\sb3$ is the only graph that has a $W\sb3$-minor and no $W\sb4$-minor. Oxley characterized the graphs that have a $W\sb4$-minor and no $W\sb5$-minor. We characterize the planar graphs that have a $W\sb5$-minor and no $W\sb6$-minor. We also determine the best-possible upper bound on the number of edges of such a graph.

Modern applications often need to manage and analyze widely diverse datasets that span multiple data models [1], [2], [3], [4], [5]. Warehousing the data through Extract-Transform-Load (ETL) processes can be expensive in such scenarios. Transforming disparate data into a single data model may degrade performance. Further, curating diverse datasets and maintaining the pipeline can prove to be labor intensive. As a result, an emerging trend is to shift the focus to federating specialized data stores and enabling query processing across heterogeneous data models [6]. This shift can bring many advantages: First, systems can natively leverage multiple data models, which can translate to maximizing the semantic expressiveness of underlying interfaces and leveraging the internal processing capabilities of component data stores. Second, federated architectures support query-specific data integration with just-in-time transformation and migration, which has the potential to significantly reduce the operational complexity and overhead. Projects that focus on developing systems in this research area stem from various backgrounds and address diverse concerns, which could make it difficult to form a consistent view of the work in this area. In this survey, we introduce a taxonomy for describing the state of the art and propose a systematic evaluation framework conducive to understanding of query-processing characteristics in the relevant systems. We use the framework to assess four representative implementations: BigDAWG [7], [8], CloudMdsQL [9], [10], Myria [11], [12], and Apache Drill [13].

For a set S of vertices and the vertex v in a connected graph G, $$\displaystyle \max _{x \in S}dx,v$$ is called the S-eccentricity of v in G. The set of vertices with minimum S-eccentricity is called the S-center of G. Any set Aof vertices of G such that A is an S-center for some set S of vertices of G is called a center set. We identify the center sets of certain classes of graphs namely, Block graphs, $$K_{m,n}$$, $$K_n-e$$, wheel graphs, odd cycles and symmetric even graphs. A graph G is called center critical if there does not a exist proper subset S of the vertex set whose S-center is the center of the graph. Here we characterize this class of graphs.

In this paper, we introduce chromatic number of line graph using α-cuts. The concept of chromatic number of fuzzy graphs wasintroduced by Munoz et.al, later Eslahchi and onagh. They are defined by the fuzzy chromatic number of complete graphs(kn),cycle graph(cn),star graph(sn),wheel graph(wn),and line graph are found and results are summarized.

An edge-colored graph G is rainbow k-connected, if for every two vertices of G, there are k internally disjoint rainbow paths, i.e., if no two edges of each path are colored the same. The minimum number of colors needed for which there exists a rainbow k-connected coloring of G, $$rc_k(G)$$ , is the rainbow k-connection number of G. Let G and H be two connected graphs, where O is an orientation of G. Let $${\vec {e}}$$ be an oriented edge of H. The edge-comb product of G (under the orientation O) and H on $$\vec {e}$$ , $$G{}^o\rhd _{\vec {e}}H$$ , is a graph obtained by taking one copy of G and |E(G)| copies of H and identifying the i-th copy of H at the edge $${\vec {e}}$$ to the i-th edge of G, where the two edges have the same orientation. In this paper, we provide sharp lower and upper bounds for rainbow 2-connection numbers of edge-comb product of a cycle and a Hamiltonian graph. We also determine the rainbow 2-connection numbers of edge-comb product of a cycle with some graphs, i.e. complete graph, fan graph, cycle graph, and wheel graph.

Subgraph Isomorphism is a fundamental problem in graph theory. It has many applications in social network analysis, molecular investigations, knowledge graphs, etc. Given a Query Graph and a Data Graph, the target of Subgraph Isomorphism, i.e., Subgraph Matching, is to determine if this Query Graph is isomorphic to any subgraph of the Data Graph. This work proposes a new type of Query Graph, combined with multiple general Query Graphs. We call it Compulsory-Optional Query Graph (CO Query Graph). This new type of Query Graph contains all the vertices in the combined general Query Graph, and each vertex corresponds to a search priority. Based on CO Query Graph, the previous multiple match processes can be reduced to one. It tremendously improves search efficiency. The Subgraph Isomorphism based on this new kind of Query Graph is an extension and improvement of the previous Subgraph Isomorphism studies. We propose a backtracking-pruning-based CO solver (BPC). This algorithm builds on the backtracking-pruning framework. BPC modifies the output criterion and matching conditions to satisfy the CO query context. A case study of real-world graph data illustrates that BPC built on CO Query Graph is more efficient than conventional Query Graphs. To verify the effectiveness of our method, we conducted experiments on the synthetic graph and real-world data. The results show that the BPC can significantly reduce the search space and improve the search efficiency in the recursive calls and the response time. Experiments resulting from synthetic graph data analysis allow us to primarily identify the critical factor that affects the efficiency of the BPC primarily.