Properties Of Graphs Research Articles

Determination and Analysis of Domination Numbers for Boundary Graph and Boundary Neighbour Graph Using MATLAB

Vertex domination is a key concept in graph theory, essential for analyzing the structural properties of graphs. This study explores the use of vertex domination to determine the domination numbers for Two specific graph structures: boundary graphs and boundary Neighbor graphs. A graph's dominance number is the minimal collection of vertex in which each vertex is simultaneously in the prevailing set and Neighboring to one of its vertex. A boundary graph is created by adding a pendant border to a circle, while the boundary graph consists of k linearly arranged paths joined at a common vertex, known as the spine. This research aims to identify the larger for each of these structures and employs MATLAB to implement the theoretical algorithms computationally. A MATLAB-based algorithm is provided to compute the domination number γ(H), identify the dominating set, and visualize the graph with highlighted dominated vertices. Computer experiments are conducted to test and validate the theoretical findings. This paper demonstrates the practical application of domination numbers using MATLAB, offering insights into the efficiency and structural properties of these graphs in both theoretical and real-world contexts.

Upper bounds for some graph invariants in terms of blocks and cut-vertices

A vertex v in a graph G is cut-vertex if removing v increases the number of connected components. Denote by cut(G) the number of cut-vertices in G. Obtaining bounds for graph parameters in terms of cut(G) is a research tradition. For graph invariants π satisfying some basic properties, we obtain upper bounds of form π(G)≤Ψ(G)=|V(G)|−(b∗−1)(cut(G)+ΔH(C)−β), where ΔH(C) is the maximum number of blocks attached to a cut-vertex, b∗ is the smallest order of a block in G and β∈{2,3} depends on the type of π. The bounds are too effective for graphs with a large number of cut-vertices. We solve an extremal problem concerning Ψ(G). These invariants include many known parameters such as the chromatic number, the coloring number and Grundy chromatic number. Finally we obtain improved bounds for the Grundy number of triangle-free graphs G in terms of cut(G). We prove the sharpness of some of the main bounds established in this paper.

Quadratic descriptors and reduction methods in a two-layered model for compound inference

Compound inference models are crucial for discovering novel drugs in bioinformatics and chemo-informatics. These models rely heavily on useful descriptors of chemical compounds that effectively capture important information about the underlying compounds for constructing accurate prediction functions. In this article, we introduce quadratic descriptors, the products of two graph-theoretic descriptors, to enhance the learning performance of a novel two-layered compound inference model. A mixed-integer linear programming formulation is designed to approximate these quadratic descriptors for inferring desired compounds with the two-layered model. Furthermore, we introduce different methods to reduce descriptors, aiming to avoid computational complexity and overfitting issues during the learning process caused by the large number of quadratic descriptors. Experimental results show that for 32 chemical properties of monomers and 10 chemical properties of polymers, the prediction functions constructed by the proposed method achieved high test coefficients of determination. Furthermore, our method inferred chemical compounds in a time ranging from a few seconds to approximately 60 s. These results indicate a strong correlation between the properties of chemical graphs and their quadratic graph-theoretic descriptors.

Topological numbers in uniform intuitionistic fuzzy environment and their application in neural network

Intuitionistic Fuzzy sets combine the ideas of uniformity, membership, and non-membership grades of the elements. Similarly, Intuitionistic Fuzzy Graphs are the generalization of simple fuzzy graphs. Depending on the uniformity of the fuzzy graphs (USIF) they can be categorized in different ways via membership values. From the idea of uniform fuzzy topological indices, we have developed the concepts of uniform intuitionistic fuzzy topological indices for uniform intuitionistic fuzzy graphs. This idea provides a more adaptable and nuanced representation of structural properties in graphs or networks. According to the theory of fuzzy topological indices, the importance of topological indices changes depending on the circumstance and the specific problem at hand. When the interactions between nodes are uncertain but not always hesitant, fuzzy graph theory and its adjusted topological indices are sufficient to capture and assess the underlying structure. In such cases where uncertainty is more complicated and hesitation is a major problem then there are better ways to address by intuitionistic fuzzy graph theory and the topological indices that go along with it. This article, developed the concept of uniform intuitionistic fuzzy graphs afresh and proposed Intuitionistic Fuzzy Topological Indices. We determine these indices using the topological indices and labeling of crisp graphs, rather than relying on the degrees of intuitionistic fuzzy graphs and edge portions. This approach is then applied to find intuitionistic fuzzy topological indices. Also, we have provided the MATLAB algorithm to illustrate the concept of IF labeling of cellular neural networks of any order. An example is given to explain the idea and approach towards one kind of uniform intuitionistic fuzzy graph represented by Cellular Neural Networks and graphical plots of the indices involved are also made.

Entropy Measures of $g-C_{3}N_{5}$ using Topological Indices E. Lavanya

Topological descriptors are non-empirical numerical quantities that characterise molecular structures. These descriptors are essential to the QSAR/QSPR approaches, as they provide theoretical chemists with a basis for investigating and synthesising chemical structures. Entropic measures are a type of topological descriptors that have many applications, ranging from the quantitative description of a chemical structure to the investigation of specific chemical properties of molecular graphs. Shannon's entropy metrics characterise graphs and networks by analysing their structural information. This study is mainly concerned with the computation of analytical expressions for degree-based entropy metrics for the chemical graph of $g-C_{3}N_{5}$. In addition to shedding light on the relationship between entropy measures and molecular structure, the numerical results of the entropy measures derived in this paper cast light on the relationship between entropy measures and molecular structure.

On topological characterizations and computational analysis of benzenoid networks for drug discovery and development.

Topological indices are numerical invariants that provide key insights into the structural properties of molecular graphs and are crucial in predicting physio-chemical and biological activities. This paper applies established computational methodologies for analyzing benzenoid networks and their application to polycyclic aromatic hydrocarbons (PAHs) through degree-based topological indices computed via M-polynomial and NM-polynomial approaches. By examining tessellations, including linear chain, hexagonal, rhomboidal, and triangular configurations alongside their line graphs, this work highlights the influence of molecular topology on biological activity. Notably, the line graph of hexagonal tessellations resembling Kagome structures exhibits the highest potential bioactivity, revealing additional connectivity patterns that offer a structured framework for early-stage drug discovery and potentially enhance the understanding of molecular interactions. These findings underscore the value of topological indices in identifying key structural features, reducing attrition rates in drug development, and improving screening technologies, contributing to efficient drug design.

On carbon nanocones CNCm [n] and their face index

Over recent decades, there has been remarkable advancement in the field of nanomaterials, particularly with the emergence of carbon nanotubes, boron nanoclusters, boron nanowires, and boron nanotubes, among others. Among these nanomaterials, carbon-based variants are particularly noteworthy due to their exceptional specific geometric and electronic properties, which make them vital in the realm of nanotechnology. The face index of a graph is a significant metric in graph theory, providing essential insights into the structural characteristics of a graph. This index explains aspects of connectivity and planarity, thereby enabling researchers to analyze and comprehend the properties of graphs more effectively. In this study, we present calculations of the recently introduced face index for the graphical structure of carbon nanotubes CNCm [n] that anticipates to assist researchers and experimentalists in developing experiments/procedures aimed at elucidating the unknown physical and chemical properties of carbon nanocones.

Topological Analysis of Fractal Binary and Ternary Trees

Fractals are complex geometric objects that seem the same at different scales and have self-similarity. Because of this special characteristic, fractals can be used to simulate intricate biological processes. Fractal binary and ternary trees are novel data types that combine the productiveness of tree architectures with the ideas of fractal geometry. In addition to self-similarity and scalability, these trees have potential across a range of computer science and medical applications. The main goal of the study is to identify and analyze topological indices and graph entropy for fractal binary and fractal ternary trees. By examining indices such as the Randić index, Zagreb indices, and entropy measurements, the study aims to obtain a comprehensive knowledge of the structural complexity and information-theoretic properties of these fractal graphs. The study starts with the vertex and edge partitioning of fractal binary and ternary trees in order to distinguish different structural classes. Using these partitions, we obtained topological indices and graph entropy values for the fractal trees. Also, this study compares the topological indices for each fractal tree with the number of copies in the fractal dimension for a succession of graphs.

High-Performance Graph Storage and Mutation for Graph Processing and Streaming

The growing need for managing extensive dynamic datasets has propelled graph processing and streaming to the forefront of the data processing community. Given the irregularity of graph workloads and the large scale of real-world graphs, researchers face numerous challenges when designing high-performance graph processing and streaming systems, due to the sheer volume, intricacy, and continual evolution of graph data. In this paper, we highlight the challenges related to two vital aspects within Graph Processing Systems that significantly impact the overall system performance: 1) the graph storage, encompassing the data structures storing vertices and edges, and 2) graph mutation protocols, referring to the ingestion and storage of new graph updates, such as additions of edges and vertices. Our paper provides a practical taxonomy of techniques designed to improve the efficiency of graph storage and mutation, by reviewing state-of-the-art systems and highlighting the challenges they face in offering a good performance tradeoff for read, write, and memory consumption. Consequently, this enables us to highlight overlooked aspects of performance, that are essential for real-world applications, such as the lack of mutation protocols for graph properties and auxiliary graph data, lack of configurability and cross-platform evaluation of solutions for graph processing and streaming.

Leader-Following Consensus of Linear Multiagent Systems With Aperiodically Sampled Outputs: A Distributed Impulsive-Observer-Based Approach.

This article studies the leader-following consensus problem for a class of linear multiagent systems over a directed graph with aperiodically sampled outputs. First, a novel distributed impulsive-observer-based consensus protocol is designed. This protocol requires only the output measurements at sporadic time instants for the observer and control gain design. Second, by using time-varying Lyapunov function techniques, sufficient conditions for exponential stability of a class of linear impulsive systems are established; subsequently, these stability results are applied for the distributed impulsive observer design. Different from the existing related works, the impulsive observer gain designed in this work is decoupled from the graph properties. As a result, once the impulsive observer gain is designed for one network topology, it can be directly used for other network topologies, as long as the graph properties and the dynamics of local agents satisfy certain conditions. Furthermore, the resilience of the designed protocol is tested under denial of service (DoS) attacks. It is shown that the protocol is robust with respect to low-frequency DoS attacks occurring in the observer communication network. Finally, two examples illustrating the validity and effectiveness of the proposed protocol are included.

Universal insulating-to-metallic crossover in tight-binding random geometric graphs

Abstract Within the scattering matrix approach to electronic transport, the scattering and transport properties of tight-binding random graphs are analyzed. In particular, we compute the scattering matrix elements, the transmission, the channel-to-channel transmission distributions (including the total transmission distribution), the shot noise power and the elastic enhancement factor. Two graph models are considered: random geometric graphs (RGGs) and bipartite RGGs. The results show an insulating to a metallic crossover in the scattering and transport properties by increasing the average degree of the graphs from small to large values. Also, the scattering and transport properties are shown to be invariant under a scaling parameter depending on the average degree and the graph size. Furthermore, for large connectivity and in the perfect coupling regime, the scattering and transport properties of both graph models are well described by the random matrix theory predictions of electronic transport, except for bipartite graphs in particular scattering setups. Our results may be verified experimentally in artificial photonic setups.

Optimization of HIV drugs through MCDM technique Analytic Hierarchy Process(AHP).

Topological indices are crucial tools for predicting the physicochemical and biological features of different drugs. They are numerical values obtained from the structure of chemical molecules. These indices, particularly the degree-based TIs are a useful tools for evaluating the connection between a compound's structure and its attributes. This study addresses the research problemof how to optimize drug design for HIV treatment using degree-based topological indices. The need for safer and more effective medicines for HIV is further emphasized by the advent of drug resistance and severe negative effects from current therapies. Employing degree-based graph invariants, the study investigates 13 HIV drugs by applying a quantitative structure-property relationship (QSPR) technique to associate their molecular structures with their physical properties. HIV drugs are ranked using the Analytic Hierarchy Process (AHP) according to specific parameters. The findings of the study demonstrate how well these approaches can determine the most effective possible drug combinations and designs, offering insightful information in developing improved HIV treatments.

Whole-brain functional connectivity and structural network properties in stroke patients with hemiplegia.

This study explored structural and functional alterations in the whole brain of stroke patients with hemiplegia. We collected multimodal magnetic resonance images of 24 patients with ischaemic stroke and 16 age-matched controls. Resting-state functional connectivity (FC) for all brain regions was evaluated. Diffusion tensor imaging was used to construct white matter structural networks, and the graph properties of the structural network were analysed using graph theory to determine group differences. The ipsilesional posterior parietal cortex (PPC) in the frontoparietal network accounts for more than half of the 25 brain regions with altered FC in stroke patients. The nodal efficiency of multiple ipsilesional frontal lobes and cerebellar regions, such as the ipsilateral cerebellum 8, was reduced. The contralesional cerebellum 8 showed elevated FC with the lingual gyrus and the visual network. Our results suggest that the PPC and cerebellum 8 are regions worthy of in-depth study. The cerebellum 8 may supplement deficits in motor balance function by enhancing functional congruence with the visual area. This study identified key brain regions and characteristics that exhibit structural and functional changes following stroke injury.

Assembly Graph as the Rosetta Stone of Ecological Assembly.

Ecological assembly-the process of ecological community formation through species introductions-has recently seen exciting theoretical advancements across dynamical, informational, and probabilistic approaches. However, these theories often remain inaccessible to non-theoreticians, and they lack a unifying lens. Here, I introduce the assembly graph as an integrative tool to connect these emerging theories. The assembly graph visually represents assembly dynamics, where nodes symbolise species combinations and edges represent transitions driven by species introductions. Through the lens of assembly graphs, I review how ecological processes reduce uncertainty in random species arrivals (informational approach), identify graphical properties that guarantee species coexistence and examine how the class of dynamical models constrain the topology of assembly graphs (dynamical approach), and quantify transition probabilities with incomplete information (probabilistic approach). To facilitate empirical testing, I also review methods to decompose complex assembly graphs into smaller, measurable components, as well as computational tools for deriving empirical assembly graphs. In sum, this math-light review of theoretical progress aims to catalyse empirical research towards a predictive understanding of ecological assembly.

Ahlfors-david Regularity of Intrinsically Quasi-symmetric Sections in Metric Spaces

We introduce a definition of intrinsically quasi-symmetric sections in metric spaces and we prove the Ahlfors-David regularity for this class of sections. We follow a recent result by Le Donne and the author where we generalize the notion of intrinsically Lipschitz graphs in the sense of Franchi, Serapioni and Serra Cassano. We do this by focusing our attention on the graph property instead of the map one.

Analyzing Pathway Energy And Time Complexity In Flower Families Through Vertex-Disjoint Paths

This research delves into the pathway energy framework for flower families, a class of simple connected graphs, whose path matrix p is constructed such that each entry pij quantifies the maximum number of vertex-disjoint paths. By analyzing the characteristic values of this matrix, we establish the pathway energy bounds specific to these flower graph families. Additionally, a comprehensive algorithm is developed to evaluate the time complexity across different flower family configurations, utilizing numerous trials to capture their average, maximum, and minimum computational behaviors. This analysis offers a comparative study of the structural intricacies that lead to increased computational complexity, highlighting which graph topologies tend to impose higher algorithmic challenges. The proposed method introduces a refined and adaptable approach, deepening the exploration of characteristic graph properties and their computational impact, thereby expanding the practical applications of these findings in graph theory.

Neighborhood version of molecular descriptors for benzene type ring graphs

This study explores the utilization of topological graph invariants, or molecular descriptors, to mathematically model chemical compounds. These descriptors are vital in quantifying the physio-chemical characteristics of a compound, and are commonly represented as shapes such as polygons, bushes, and grapes. Specifically, this research focuses on computing selected fifth multiplicative first and second Zagreb indices, third and fourth multiplicative general fifth multiplicative Zagreb indices, and other degree-based topological indices for the benzene ring graph ([Formula: see text] and the simple bounded dual of benzene ring graph ([Formula: see text]. The findings of this study are then compared to demonstrate the impact of these molecular descriptors.

Novel M-Polynomials for Computing Graph Invariants of Certain Drugs: Targeting COVID-19 and Omicron

Novel M-Polynomials for Computing Graph Invariants of Certain Drugs: Targeting COVID-19 and Omicron

SOME TOPOLOGICAL INDICES VALUES OF THE CONVEX POLYTOPE Dn AND ITS SUBDIVISION GRAPH S(Dn)

Graph theory has been studied different areas such as computer science, information, mathematics, natural sciences, and so on. Especially, it has been the most important mathematical analysis tools for the study the analysis of chemistry science. For example, the studies of descriptors in quantitative structure property relationship (QSPR) and quantitative structure activity relationship (QSAR) studies in the chemistry science have been used graph theory tecniques. Furtermore, topological indices have been also used to determine combinatorial properties of chemical graphs. In this paper, exact values of some eccentricity-based topological indices of the convex polytope Dn and its subdivision graph S(Dn) have been determined.

The harmonic index and some Hamiltonian properties of graphs

The harmonic index and some Hamiltonian properties of graphs