Properties Of Graphs Research Articles

EQUIPRIME FUZZY GRAPH OF A NEARRING WITH RESPECT TO A LEVEL IDEAL

In this paper, we introduce an equiprime fuzzy graph of a nearring with respect to the level ideal of a fuzzy ideal. We interrelate graph theoretical properties of the graph and ideal theoretical properties of nearring. We show that the properties like vertex cut, connectedness of the graph depend on the properties of the fuzzy ideal. We define ideal symmetry of the graph and find conditions for the graph to be ideal symmetric. If the fuzzy ideal is equiprime then we show that the level set induces a fuzzy clique. We find conditions required for the level set to be the vertex cover of the graph. We find interrelation between equiprime fuzzy graph and fuzzy graph of nearring with respect to level ideal. We study properties of the graph under neaaring homomorphism. We prove that the connectedness of the graph in homomorphic image depends on properties of ideal. We obtain conditions required for homomorphic image of an equiprime fuzzy ideal to be an equiprime fuzzy ideal.

Спектральные инварианты графов и их приложения в комбинаторике

In this research, we present recent results obtained by the authors. They related to spectral invariants of graphs admitting an arbitrary large cyclic group action. To illustrate them we use the family of circulant graphs = 𝐶𝑛(𝑠1, 𝑠2, . . . , ).The Chebyshev polynomials provide a significant analytical tools for studying theproperties of such graphs and their characteristic polynomials. In particular, this gives a way to find analytical expressions for the number of spanning trees (𝑛), the number of rooted spanning forests 𝑓𝐺(𝑛) and the Kirchhoff index (𝐺𝑛) of a graph. We are interested in the behaviour of these invariants for sufficiently large 𝑛. We provideasymptotic formulas of the above mentioned invariants. These results were motivated by problems arising in theoretical physics, biology and chemistry.

The Largest Eigenvalue of the Signless Laplacian and Some Hamiltonian Properties of Graphs

The Largest Eigenvalue of the Signless Laplacian and Some Hamiltonian Properties of Graphs

On the Spectral Properties of Graphs with Rank 4

On the Spectral Properties of Graphs with Rank 4

Mathematical aspects and topological properties of two chemical networks

<abstract><p>Graphs give a mathematical model of molecules, and thery are used extensively in chemical investigation. Strategically selections of graph invariants (formerly called "topological indices" or "molecular descriptors") are used in the mathematical modeling of the physio-chemical, pharmacologic, toxicological, and other aspects of chemical compounds. This paper describes a new technique to compute topological indices of two types of chemical networks. Our research examines the mathematical characteristics of molecular descriptors, particularly those that depend on graph degrees. We derive a compact mathematical analysis and neighborhood multiplicative topological indices for product of graphs ($ \mathcal{L} $) and tetrahedral diamond lattices ($ \Omega $). In this paper, the fifth multiplicative Zagreb index, the general fifth multiplicative Zagreb index, the fifth multiplicative hyper-Zagreb index, the fifth multiplicative product connectivity index, the fifth multiplicative sum connectivity index, the fifth multiplicative geometric-arithmetic index, the fifth multiplicative harmonic index and the fifth multiplicative redefined Zagreb index are determined. The comparison study of these topological indices is also discussed.</p></abstract>

On the application of some graph invariants in communication network research tasks

The problem of checking the possible isomorphism of graphs has a wide practical application and is an important problem for theoretical computer science in general and the theory of algorithms in particular. Among the numerous areas of application of algorithms for solving the problem of determining graph isomorphism, we note the problem of syntactic and structural pattern recognition, some problems of mathematical chemistry and chemoinformatics (study of molecular structures of chemical compounds), problems related to the study of social networks (for example, linking several accounts of one user on Facebook). It is not known whether it belongs to a polynomial algorithm for this problem — assuming P≠NP. It is known, for example, that NP-complete is the related problem of finding an isomorphic subgraph in a given graph. Thus, the current research is relevant, which is aimed at solving the problem of checking isomorphism for both arbitrary graphs and graphs of a special type (in practice, both exact and heuristic algorithms can be used for such studies). In various algorithms for working with graphs, one of the most common invariants is the vector of degrees. However, the use of this invariant alone for constructing most practical algorithms on graphs is apparently not sufficient; its possible generalization is the more complex invariant considered by the authors — the vector of second–order degrees. At the same time, the graphs considered in this paper with the generated vector of second-order degrees can be considered models for many real complex problems. Previously, works were published in which the orders of application of invariants calculated in polynomial time were analyzed, and such variants of algorithms for which small degrees of the applied polynomial are needed. When analyzing such algorithms, there are problems of comparing the invariants under consideration — comparing by some specially selected metric that reflects the «quality» of the invariant on the subset of the set of all graphs under consideration. The article shows that when using any natural metric, the vector of second-order degrees is better than the widely used Randich index.

Topological Aspects of Dendrimers via Connection-Based Descriptors

Topological indices (TIs) have been practiced for distinct wide-ranging physicochemical applications, especially used to characterize and model the chemical structures of various molecular compounds such as dendrimers, nanotubes and neural networks with respect to their certain properties such as solubility, chemical stability and low cytotoxicity. Dendrimers are prolonged artificially synthesized or amalgamated natural macromolecules with a sequential layer of branches enclosing a central core. A present-day trend in mathematical and computational chemistry is the characterization of molecular structure by applying topological approaches, including numerical graph invariants. Among topological descriptors, Zagreb connection indices (ZCIs) have much importance. This manuscript involves the establishment of general results to calculate ZCIs, namely first ZCI (FZCI), second ZCI (SZCI), third ZCI (TZCI), modified FZCI, modified SZCI and modified TZCI of two special types of dendrimers nanostars, namely, poly propylene imine octamin (PPIO) dendrimer and poly (propyl) ether imine (PPEtIm) dendrimer. Furthermore, we provide the numerical and graphical comparative analysis of our calculated results for both types of dendrimers with each other.

О МАШИННОМ РЕШЕНИИ ЗАДАЧ В АЛГЕБРЕ ГРАФОВ

The article discusses the method of using the package of mathematical symbolic calculations Maple in the study of the university course "Graph Theory". The features of using the with (Graph Theory) sub package are considered. The technique of computer research and finding the main parameters of the graph is shown. It is noted that for the machine solution of the problem of isomorphism of two graphs in addition to the commands of the sub package with (GraphTheory) the commands of the sub package with (LinearAlgebra) can be used. When checking whether the grapes are isomorphic or not some questions naturally rise about the structure of the graph and its parameters which preserved under isomorphism. The final verification of the connection of the conjugacy matrices of the graphs being tested is carried out only after the incomplete graph invariants have been successfully verified.

Mathematical Model and Simulation of Blood Flow Dynamics in Renal Interlobar Artery of Patients with Human immunodeficiency Virus

Mathematical models were developed considering wall movement, blood pulsation and flow dynamics of the blood in the interlobar artery. The formulated models were based on the fact that the motion of blood vessel wall is not only influenced by pulsation, but also other physiological processes like heartbeat, breathing, and body posture. The Newton’s second Law of motion was employed in assembling the forces acting on the interlobar artery. The wall shear stress (WSS) was studied alongside, arterial walls to study the actual flow dynamics and investigate the blood flow behaviour. The results of the study were presented on both two – dimensional (2D) and three (3D) – dimensional graphs showing a more realistic interaction between the arterial wall and the blood flow in patients with HIV/AIDS. It was deduced that the blood flow velocity decreased with time across the varying frequency from 0.20Hz to 0.50Hz in the interlobar arterial channel.

A novel analysis of the time-fractional nonlinear dispersive K(m, n, 1) equations using the homotopy perturbation transform method and Yang transform decomposition method

<abstract><p>The main features of scientific effort in physics and engineering are the development of models for various physical issues and the development of solutions. In this paper, we investigate the numerical solution of time-fractional non-linear dispersive K(m, n, 1) type equations using two innovative approaches: the homotopy perturbation transform method and Yang transform decomposition method. Our suggested approaches elegantly combine Yang transform, homotopy perturbation method (HPM) and adomian decomposition method (ADM). With the help of the Yang transform, we first convert the problem into its differential partner before using HPM to get the He's polynomials and ADM to get the Adomian polynomials, both of which are extremely effective supports for non-linear issues. In this case, Caputo sense is used for defining the fractional derivative. The derived solutions are shown in series form and converge quickly. To ensure the effectiveness and applicability of the proposed approaches, the examined problems were analyzed using various fractional orders. We analyze and demonstrate the validity and applicability of the solution approaches under consideration with given initial conditions. Two and three dimensional graphs reflect the outcomes that were attained. To verify the effectiveness of the strategies, numerical simulations are presented. The numerical outcomes demonstrate that only a small number of terms are required to arrive at an approximation that is exact, efficient, and trustworthy. The results of this study demonstrate that the studied methods are effective and strong in solving nonlinear differential equations that appear in science and technology.</p></abstract>

Approach to reduce the communication cost when partitioning a big graph

While several graph mining systems have been developed to run different graph mining algorithms on such large networks, they have difficulty processing Web-scale graphs owing to the significant communication and I/O expenses incurred when workers connect with one another and frequently read subgraphs. In this study, we provide a cost-effective, scalable, distributed method for mining real-world graphs by exploiting graph properties. Our technique drastically decreases the communication cost, the main bottleneck of distributed systems, by using alternate edge placement criteria depending on the types of vertices.

Enhanced Clustering Based OSN Privacy Preservation to Ensure k-Anonymity, t-Closeness, l-Diversity, and Balanced Privacy Utility

Online Social Networks (OSN) sites allow end-users to share a great deal of information, which may also contain sensitive information, that may be subject to commercial or non-commercial privacy attacks. As a result, guaranteeing various levels of privacy is critical while publishing data by OSNs. The clustering-based solutions proved an effective mechanism to achieve the privacy notions in OSNs. But fixed clustering limits the performance and scalability. Data utility degrades with increased privacy, so balancing the privacy utility trade-off is an open research issue. The research has proposed a novel privacy preservation model using the enhanced clustering mechanism to overcome this issue. The proposed model includes phases like pre-processing, enhanced clustering, and ensuring privacy preservation. The enhanced clustering algorithm is the second phase where authors modified the existing fixed k-means clustering using the threshold approach. The threshold value is determined based on the supplied OSN data of edges, nodes, and user attributes. Clusters are k-anonymized with multiple graph properties by a novel one-pass algorithm. After achieving the k-anonymity of clusters, optimization was performed to achieve all privacy models, such as k-anonymity, t-closeness, and l-diversity. The proposed privacy framework achieves privacy of all three network components, i.e., link, node, and user attributes, with improved utility. The authors compare the proposed technique to underlying methods using OSN Yelp and Facebook datasets. The proposed approach outperformed the underlying state of art methods for Degree of Anonymization, computational efficiency, and information loss.

Simple-intersection graphs of rings

<abstract><p>Let $ R $ be a ring with unity. In this paper, we introduce a new graph associated with $ R $ called the simple-intersection graph of $ R $, denoted by $ GS(R) $. The vertices of $ GS(R) $ are the nonzero ideals of $ R $, and two vertices are adjacent if and only if their intersection is a nonzero simple ideal. We study the interplay between the algebraic properties of $ R $, and the graph properties of $ GS(R) $ such as connectedness, bipartiteness, girth, dominating sets, etc. Moreover, we determine the precise values of the girth and diameter of $ GS(R) $, as well as give a formula to compute the clique and domination numbers of $ GS(R) $.</p></abstract>

Semi-random process without replacement

Semi-random processes involve an adaptive decision-maker, whose goal is to achieve some pre-determined objective in an online randomized environment. We introduce and study a semi-random multigraph process, which forms a no-replacement variant of the process that was introduced by Ben-Eliezer, Hefetz, Kronenberg, Parczyk, Shikhelman and Stojakovi\'c (2020). The process starts with an empty graph on the vertex set $[n]$. For every positive integers $q$ and $1\leq r\leq n$, in the $((q-1)n+r)$th round of the process, the decision-maker, called \emph{Builder}, is offered the vertex $\pi_q(r)$, where $\pi_1, \pi_2, \ldots$ is a sequence of permutations in $S_n$, chosen independently and uniformly at random. Builder then chooses an additional vertex (according to a strategy of his choice) and connects it by an edge to $\pi_q(r)$. For several natural graph properties, such as $k$-connectivity, minimum degree at least $k$, and building a given spanning graph (labeled or unlabeled), we determine the typical number of rounds Builder needs in order to construct a graph having the desired property. Along the way we introduce and analyze an urn model which may also have independent interest.

On Vertex-Degree-Based Indices of Monogenic Semigroup Graphs

Albertson and the reduced Sombor indices are vertex-degree-based graph invariants that given in [5] and [18], defined as
 
 Alb(G)=\sum_{uv\in E(G)}\left|d_{u}-d_{v}\right|, SO_{red}(G)=\sum_{uv\in E(G)}\sqrt{(d_{u}-1)^{2}+(d_{v}-1)^{2}},
 
 respectively. 
 
 In this work we show that a calculation of Albertson and reduced Sombor index which are vertex-degree-based topological indices, over monogenic semigroup graphs.

An Algorithmic Meta-Theorem for Graph Modification to Planarity and FOL

In general, a graph modification problem is defined by a graph modification operation ⊠ and a target graph property 𝒫. Typically, the modification operation ⊠ may be vertex deletion , edge deletion , edge contraction , or edge addition and the question is, given a graph G and an integer k , whether it is possible to transform G to a graph in 𝒫 after applying the operation ⊠ k times on G . This problem has been extensively studied for particular instantiations of ⊠ and 𝒫. In this article, we consider the general property 𝒫 𝛗 of being planar and, additionally, being a model of some First-Order Logic (FOL) sentence 𝛗 (an FOL-sentence). We call the corresponding meta-problem Graph ⊠-Modification to Planarity and 𝛗 and prove the following algorithmic meta-theorem: there exists a function f : ℕ 2 → ℕ such that, for every ⊠ and every FOL-sentence 𝛗, the Graph ⊠-Modification to Planarity and 𝛗 is solvable in f ( k,|𝛗| )⋅ n 2 time. The proof constitutes a hybrid of two different classic techniques in graph algorithms. The first is the irrelevant vertex technique that is typically used in the context of Graph Minors and deals with properties such as planarity or surface-embeddability (that are not FOL-expressible) and the second is the use of Gaifman’s locality theorem that is the theoretical base for the meta-algorithmic study of FOL-expressible problems.

매듭사영의 BLL 그래프의 정의와 그 응용에 대한 연구

This paper is the research on the Knot theory in Topology. A knot is a simple closed curve in ℝ and its projection onto a plane in ℝ is called a knot projection. As the results of this paper we define a BLL(Bidirectional Linear Link) graph for a knot projection which is a bidirectional linear link representing the relations between arcs of a knot projection and obtain some properties of the BLL graphs. We also define an Eulerian cycle of the BLL graph and an Eulerian cycle of a knot projection. As the main results of this paper, we obtain the equivalent conditions of being an alternation knot projection as follows: (1) an out-degree of every vertex of the corresponding BLL graph is 2; (2) the corresponding BLL graph has an Eulerian cycle; (3) the knot projection has an Eulerian cycle. As the subsequent study, using these results of the BLL graphs, we propose the analysis on the BLL graphs for deformation operation obtaining a new alternating knot projection, decision on the tricolorability of a knot projection, and a polynomial of a knot projection.

Minimal induced subgraphs of the class of 2-connected non-Hamiltonian wheel-free graphs

Given a graph G and a graph property P we say that G is minimal with respect to P if no proper induced subgraph of G has the property P. An HC-obstruction is a minimal 2-connected non-Hamiltonian graph. Given a graph H, a graph G is H-free if G has no induced subgraph isomorphic to H. The main motivation for this paper originates from a theorem of Duffus, Gould, and Jacobson (1981), which characterizes all the minimal connected graphs with no Hamiltonian path. In 1998, Brousek characterized all the claw-free HC-obstructions. On a similar note, Chiba and Furuya (2021), characterized all (not only the minimal) 2-connected non-Hamiltonian {K1,3,N3,1,1}-free graphs. Recently, Cheriyan, Hajebi, and two of us (2022), characterized all triangle-free HC-obstructions and all the HC-obstructions which are split graphs. A wheel is a graph obtained from a cycle by adding a new vertex with at least three neighbors in the cycle. In this paper we characterize all the HC-obstructions which are wheel-free graphs.

A General Computational Approach for Counting Labeled Graphs

This paper presents a general recursive formula to estimate the number of labeled graphs as well as details to evaluate the formula for the following graph properties: number of edges (graph density), degree sequence, degree distribution, classification mixing, and degree mixing, i.e., the formula estimates the number of labeled graphs that have given values for graph properties. The proposed approach can be extended to additional graph properties (e.g., number of triangles) as well as properties of bipartite graphs. For special settings in which formulas exist from previous research, simulation studies demonstrate the validity of the proposed approach. In addition, we demonstrate how our approach can be used to quantify the level of variability in values of a graph property in the subset of graphs that hold a specified value of a different graph property (or properties) constant.

Substituent effect on the photoinduced geometrical changes of Cu[formula omitted]Phen[formula omitted] complexes

This paper analyzes the substituent effect (SE) on the photo-physical process of bis-1,10-phenanthroline copper(I) complexes to provide information that may eventually be useful in designing molecular machines based on this metal. This metal complex’s ground and excited states were calculated, including the vertical, adiabatic, and intersystem crossing points (ISCP). Ab Initio Molecular Dynamics was used to find the time when the ISCP appears. The integrated properties of electron density allowed to detection of SE on the metal-ligand electron transfer and the dihedral angle between the ligands. The properties of the atomic graph permitted the analysis of SE on the equatorial coordination sites.