Properties Of Graphs Research Articles

Fault Tolerance and Node Mapping Algorithm Hierarchical Graph HCN

Embedding in an interconnection network maps the interconnection network G to H and analyzes the relationship between G and H. Hypercubes and toruses are widely known as interconnection networks, and various algorithms have been developed. The HCN graph is a network with a hierarchical structure to improve the network cost of hypercube. In this study, we analyze the fault tolerance and embedding properties of HCN graphs. As a result of the research, the HCN graph has the same node connectivity and degree, so it has the maximum fault tolerance property. In addition, an algorithm that can one-to-one node map the HCN structure to the torus structure was proposed. The embedding results showed that it was possible to embed the torus structure in the HCN graph at an extension rate of 3 and dilation of 1. The results of the embedding study mean that various algorithms developed in Torus can be efficiently used in the HCN structure.

The effective enhancement of information in 3D small-world networks of biological neuronal cells

The cardiovascular system, the kidney, or the brain, are examples of complex systems - where the properties of the systems arise because of the layout of cells in those systems. One way to characterize systems is using networks, where elements and interactions of the systems are represented as nodes and links of a graph. Network’s topology can be, in turn, measured by the small-world coefficient. Small world networks feature increased clustering and shorter paths compared to random or periodic networks of the same size. This suggests that systems with small world attributes can also efficiently transport signals, nutrients, or information within their body. While several reports in literature have illustrated that real biological systems are small-world, yet little is known about how information varies as a function of the small-world-ness () of three dimensional graphs. Here, we used a model of the brain to estimate quantitatively the information processed in 3D networks. In the model, nodes of the graph are neuronal units capable to receive, integrate and transmit signals to other neurons of the system in parallel. The information encoded in the signals was then extracted using the techniques of information theory. In simulations where the topology of networks of nodes was varied over large intervals, we found that in the range information scales linearly with the small world coefficient, with a five-fold increase. Results of the paper and review of the existing literature on model organisms suggest that a small-world architecture may offer an evolutionary advantage to biological systems.

AB/Push-Pull method for distributed optimization in time-varying directed networks

In this paper, we study the distributed optimization problem for a system of agents embedded in time-varying directed communication networks. Each agent has its own cost function and agents cooperate to determine the global decision that minimizes the summation of all individual cost functions. We consider the so-called push-pull gradient-based algorithm (termed as AB/Push-Pull) which employs both row- and column-stochastic weights simultaneously to track the optimal decision and the gradient of the global cost while ensuring consensus and optimality. We show that the algorithm converges linearly to the optimal solution over a time-varying directed network for a constant stepsize when the agent's cost function is smooth and strongly convex. The linear convergence of the method has been shown in [F. Saadatniaki, R. Xin, and U.A. Khan, Decentralized optimization over time-varying directed graphs with row and column-stochastic matrices, IEEE Trans. Autom. Control 65(11) (2020), pp. 4769–4780], where the multi-step consensus contraction parameters for row- and column-stochastic mixing matrices are not directly related to the underlying graph structure, and the explicit range for the stepsize value is not provided. With respect to [F. Saadatniaki, R. Xin, and U.A. Khan, Decentralized optimization over time-varying directed graphs with row and column-stochastic matrices, IEEE Trans. Autom. Control 65(11) (2020), pp. 4769–4780], the novelty of this work is twofold: (1) we establish the one-step consensus contraction for both row- and column-stochastic mixing matrices with the contraction parameters given explicitly in terms of the graph diameter and other graph properties; and (2) we provide explicit upper bounds for the stepsize value in terms of the properties of the cost functions, the mixing matrices, and the graph connectivity structure.

Vertex-Based Resolvability Parameters for Identification of Certain Chemical Structures.

Chemical graph theory explores chemical phenomena and entities through the conceptual framework of graph theory. In chemical graph theory, molecular structures are represented by chemical graphs, where edges and vertices correspond to bonds and atoms, respectively. Chemical graphs serve as fundamental data types in cheminformatics for illustrating chemical structures. The computable properties of graphs form the basis for quantitative structure-property and structure-activity predictions, which are central to cheminformatics. These graphs capture the physical characteristics of molecules and can be further reduced to graph-theoretical indices or descriptors. One extensively studied distance-based graph descriptor is the resolving set Z, which enables the distinction of every pair of distinct vertices in a connected simple graph. Resolving sets were specifically employed in pharmaceutical research to find patterns shared by several different drugs. Since very early times, medicinal drugs have played a significant part in human civilization. In this article, we investigate minimum resolving sets for certain significant drug molecular structures, namely, suramin (S86) and acemannan (A116).

DAEMDA: A Method with Dual-Channel Attention Encoding for miRNA-Disease Association Prediction.

A growing number of studies have shown that aberrant microRNA (miRNA) expression is closely associated with the evolution and development of various complex human diseases. These key biomarkers' identification and observation are significant for gaining a deeper understanding of disease pathogenesis and therapeutic mechanisms. Consequently, pinpointing potential miRNA-disease associations (MDA) has become a prominent bioinformatics subject, encouraging several new computational methods given the advances in graph neural networks (GNN). Nevertheless, these existing methods commonly fail to exploit the network nodes' global feature information, leaving the generation of high-quality embedding representations using graph properties as a critical unsolved issue. Addressing these challenges, we introduce the DAEMDA, a computational method designed to optimize the current models' efficacy. First, we construct similarity and heterogeneous networks involving miRNAs and diseases, relying on experimentally corroborated miRNA-disease association data and analogous information. Then, a newly-fashioned parallel dual-channel feature encoder, designed to better comprehend the global information within the heterogeneous network and generate varying embedding representations, follows this. Ultimately, employing a neural network classifier, we merge the dual-channel embedding representations and undertake association predictions between miRNA and disease nodes. The experimental results of five-fold cross-validation and case studies of major diseases based on the HMDD v3.2 database show that this method can generate high-quality embedded representations and effectively improve the accuracy of MDA prediction.

Spectral Conditions, Degree Sequences, and Graphical Properties

Integrity, tenacity, binding number, and toughness are significant parameters with which to evaluate network vulnerability and stability. However, we hardly use the definitions of these parameters to evaluate directly. According to the methods, concerning the spectral radius, we show sufficient conditions for a graph to be k-integral, k-tenacious, k-binding, and k-tough, respectively. In this way, the vulnerability and stability of networks can be easier to characterize in the future.

A Pure Mathematical Proof of 4-Colour Theorem

This work "A Pure Mathematical Proof of the 4-Colour Theorem" is related to the previous proof assisted by computer. "Triangulations of Euler Convex Polygon" provides a fresh beginning point for the proof. The central concept is to discover an extended invariance property of Standard Graph’s boundary, which is described as "3-Colour All Phase States (3CP)" in this work and it is demonstrated that the Standard Graph’s boundary and sub-bound are 3CP and 4-colorable (4-3CP) via the Expanded Operation e(+, pi) and e(-, pi). It's exciting that this regularity was discovered for the first time and the 4-3CP invariance can naturally derive the 4-Colour Theorem. The majority of the definitions, theorems and proof strategies are shown in this work.

Retracted: On Chemical Invariants of Semitotal-Point Graph and Its Line Structure of the Acyclic Kragujevac Network: A Novel Mathematical Analysis

Retracted: On Chemical Invariants of Semitotal-Point Graph and Its Line Structure of the Acyclic Kragujevac Network: A Novel Mathematical Analysis

Reciprocal degree distance and Hamiltonian properties of graphs

For a connected graph G, the reciprocal degree distance (RDD) of the graph G is defined as; RDD(G)=∑u≠vdegG⁡(u)+degG⁡(v)dG(u,v), where degG⁡(u) is the degree of u in G and dG(u,v) is the distance between u and v in G. In this article, we give sufficient condition for a graph to be traceable, Hamiltonian, Hamiltonian-connected in terms of RDD(G).

On the Distance Spectrum and Distance-Based Topological Indices of Central Vertex-Edge Join of Three Graphs

In this paper, we introduce a new graph operation based on a central graph called central vertex-edge join (denoted by $G_{n_1}^C \triangleright (G_{n_2}^V\cup G_{n_3}^E)$) and then determine the distance spectrum of $G_{n_1}^C \triangleright (G_{n_2}^V\cup G_{n_3}^E)$ in terms of the adjacency spectra of regular graphs $G_1$, $G_2$ and $G_3$ when $G_1$ is triangle-free. As a consequence of this result, we construct new families of non-D-cospectral D-equienergetic graphs. Moreover, we determine bounds for the distance spectral radius and distance energy of the central vertex-edge join of three regular graphs. In addition, we provide its results related to graph invariants like eccentric-connectivity index, connective eccentricity index, total-eccentricity index, average eccentricity index, Zagreb eccentricity indices, eccentric geometric-arithmetic index, eccentric atom-bond connectivity index, Wiener index. Using these results, we calculate the topological indices of the organic compounds Methylcyclobutane $(C_5H_{10})$ and Spirohexane $(C_6H_{10})$.

The study of line graphs of subdivision graphs of some rooted product graphs via K-Banhatti indices

The degree-based topological indices are numerical graph invariants that are used to link a molecule’s structural characteristics to its physical, and chemical characteristics. In the investigation, and study of the structural features of a chemical network, it has emerged as one of the most potent mathematical techniques. In this paper, we study the degree-based topological invariants, called K-Banhatti indices, of the line graphs of some rooted product graphs namely, [Formula: see text], [Formula: see text], and ith vertex rooted product graph [Formula: see text] which are derived by the concept of subdivision.

Interaction graph-based characterization of quantum benchmarks for improving quantum circuit mapping techniques

To execute quantum circuits on a quantum processor, they must be modified to meet the physical constraints of the quantum device. This process, called quantum circuit mapping, results in a gate/circuit depth overhead that depends on both the circuit properties and the hardware constraints, being the limited qubit connectivity a crucial restriction. In this paper, we propose to extend the characterization of quantum circuits by including qubit interaction graph properties using graph theory-based metrics in addition to previously used circuit-describing parameters. This approach allows for an in-depth analysis and clustering of quantum circuits and a comparison of performance when run on different quantum processors, aiding in developing better mapping techniques. Our study reveals a correlation between interaction graph-based parameters and mapping performance metrics for various existing configurations of quantum devices. We also provide a comprehensive collection of quantum circuits and algorithms for benchmarking future compilation techniques and quantum devices.

Evaluation of Various Topological Indices of Flabellum Graphs

Graph theory serves as an engaging arena for the investigation of proof methods within the field of discrete mathematics, and its findings find practical utility in numerous scientific domains. Chemical graph theory is a specialized branch of mathematics that uses graphs to represent and analyze the structure and properties of chemical compounds. Topological indices are mathematical properties of graphs that play a crucial role in chemistry. They provide a unique way to connect the structural characteristics of chemical compounds to their corresponding molecular graphs. The flabellum graph Fn(k,j) is obtained with the help of k≥2 duplicates of the cycle graph Cn with a common vertex (known as, central vertex). Then, in j of these duplicates, additional edges are added, joining the central vertex to all non-adjacent vertices. In this article, we compute different degree-based topological indices for flabellum graphs, including some well known indices, such as the Randić index, the atom bond connectivity index, the geometric–arithmetic index, and the Zagreb indices. This research provides an in-depth examination of these specific indices within the context of flabellum graphs. Moreover, the behavior of these indices is shown graphically, in terms of the parameters j,k, and n. Additionally, we have extended the concept of the first Zagreb index, to address the issue of cybercrime. This application enables us to identify criminals who exhibit higher levels of activity and engagement in multiple criminal activities when compared to their counterparts. Furthermore, we conducted a comprehensive comparative analysis of the first Zagreb index against the closeness centrality measure. This analysis sheds light on the effectiveness and relevance of the topological index in the context of cybercrime detection and network analysis.

Hereditary classes of ordered sets of width at most two

This paper is a contribution to the study of hereditary classes of relational structures, these classes being quasi-ordered by embeddability. It deals with the specific case of ordered sets of width two and the corresponding bichains and incomparability graphs.Several open problems about hereditary classes of relational structures which have been considered over the years have a positive answer in this case. For example, well-quasi-ordered hereditary classes of finite bipartite permutation graphs, respectively finite 321-avoiding permutations, have been characterized by Korpelainen, Lozin and Mayhill, respectively by Albert, Brignall, Ruškuc and Vatter.In this paper we present an overview of properties of these hereditary classes in the framework of the Theory of Relations as presented by Roland Fraïssé.We provide another proof of the results mentioned above. It is based on the existence of a countable universal poset of width two, obtained by the first author in 1978, his notion of multichainability (1978) (a kind of analog to letter-graphs), and metric properties of incomparability graphs. Using Laver’s theorem (1971) on better-quasi-ordering (bqo) of countable chains we prove that a wqo hereditary class of finite or countable bipartite permutation graphs is necessarily bqo. This gives a positive answer to a conjecture of Nash-Williams (1965) in this case. We extend a previous result of Albert et al. by proving that if a hereditary class of finite, respectively countable, bipartite permutation graphs is wqo, respectively bqo, then the corresponding hereditary classes of posets of width at most two and bichains are wqo, respectively bqo.Several notions of labelled wqo are also considered. We prove that they are all equivalent in the case of bipartite permutation graphs, posets of width at most two and the corresponding bichains. We characterize hereditary classes of finite bipartite permutation graphs which remain wqo when labels from a wqo are added. Hereditary classes of posets of width two, bipartite permutation graphs and the corresponding bichains having finitely many bounds are also characterized.We prove that a hereditary class of finite bipartite permutation graphs is not wqo if and only if it embeds the poset of finite subsets of N ordered by set inclusion. This answers a long standing conjecture of the first author in the case of bipartite permutation graphs.

Solution to a general version of a degree sequence variant of the Erdős–Sós conjecture

Let P denote a graph property, such as “being k-connected”, “being hamiltonian”, or “containing H as a subgraph”, etc. A graphic sequence π is potentially P-graphic if there is a realization of π having the property P. In 1991, Erdős et al. introduced the following problem: Determine the minimum even integer σ(P,n) such that every n-term graphic sequence with sum at least σ(P,n) is potentially P-graphic. The parameter σ(P,n) is known as the potential function of P, and can be viewed as a degree sequence variant of the classical extremal function ex(P,n). A simple graph G is an ℓ-tree if G=Kℓ+1, or G has a vertex whose neighborhood is a clique of order ℓ, and G−v is an ℓ-tree. Clearly, 1-trees are the usual trees. For ℓ≥1 and k≥ℓ+1, let P(ℓ,k) be the graph property “containing every ℓ-tree on k vertices as a subgraph”. For ℓ=1, Yin and Li determined σ(P(1,k),n) for k≥2 and n≥92k2+192k. This is a degree sequence variant of the Erdős–Sós conjecture. For ℓ=2, Zeng et al. determined σ(P(2,k),n) for k≥3 and n sufficiently large. In this paper, we consider the most general case ℓ≥3, and determine σ(P(ℓ,k),n) for ℓ≥3, k≥ℓ+1 and n sufficiently large.

Characterization of expansion-related properties of modular graphs

Characterization of expansion-related properties of modular graphs

Wiener index and vulnerability parameters of graphs

Let G be a finite simple graph with vertex set V(G). We define the Wiener index of G as W(G)=∑{u,v}∈V(G)d(u,v), where d(u,v) is the distance between u and v. This topological index was defined by Wiener in 1947 for studying the structural graphs of molecules (Wiener, 1947 [17]) (although, Wiener’s definition included a 1/2 multiple in front of the sum). Since that time, many theoretical results have been derived. The results we present concern the graph parameters integrity, toughness, tenacity and binding number. Lower bounds on these parameters have been used extensively to determine properties of graphs. For each of the four parameters, we provide a best possible upper bound on the Wiener index of G that guarantees a specified lower bound on that parameter. This provides a continuation of results provided in Feng et al. (2017).

SOME PROPERTIES ON COPRIME GRAPH OF GENERALIZED QUATERNION GROUPS

A coprime graph is a representation of finite groups on graphs by defining the vertex graph as an element in a group and two vertices adjacent to each other's if and only if the order of the two elements is coprime. In this research, we discuss the generalized Quaternion group and its properties. Then we discuss the properties of the coprime graph over the generalized Quaternion group by looking at its Eulerian, Hamiltonian, and Planarity sides. In general, the coprime graphs of the generalized quaternion group are not Eulerian, not Hamilton, and not planar graphs. The coprime graph of the generalized quaternion group is a planar graph if for a natural number .

THE POWER GRAPH REPRESENTATION FOR INTEGER MODULO GROUP WITH POWER PRIME ORDER

There are many applications of graphs in various fields. Starting from chemical problems, such as the molecular shape of a compound to internet network problems, we can also use graphs to depict the abstract concept of a mathematical structure.. Groups in Algebra can be represented as a graph. This is interesting because Groups are abstract objects in mathematics. The graph of a group shows the physical form of the group by looking at the relationship between its elements. So, we can know the distance of the elements. In 2013, Abawajy et al. conducted studies related to power graphs. Power graph representation of groups of integers modulo with the order of prime numbers has been carried out in 2022 by Syechah, et al. In this article, the author provides the properties of a power graph on a group of integers modulo with the order of powers of prime numbers.

On the Super (Edge)-Connectivity of Generalized Johnson Graphs

Let [Formula: see text], [Formula: see text] and [Formula: see text] be non-negative integers. The generalized Johnson graph [Formula: see text] is the graph whose vertices are the [Formula: see text]-subsets of the set [Formula: see text], and two vertices are adjacent if and only if they intersect with [Formula: see text] elements. Special cases of generalized Johnson graph include the Kneser graph [Formula: see text] and the Johnson graph [Formula: see text]. These graphs play an important role in coding theory, Ramsey theory, combinatorial geometry and hypergraphs theory. In this paper, we discuss the connectivity properties of the Kneser graph [Formula: see text] and [Formula: see text] by their symmetric properties. Specifically, with the help of some properties of vertex/edge-transitive graphs we prove that [Formula: see text] [Formula: see text] and [Formula: see text] [Formula: see text] are super restricted edge-connected. Besides, we obtain the exact value of the restricted edge-connectivity and the cyclic edge-connectivity of the Kneser graph [Formula: see text] [Formula: see text] and [Formula: see text] [Formula: see text], and further show that the Kneser graph [Formula: see text] [Formula: see text] is super vertex-connected and super cyclically edge-connected.