Cayley Picture Fuzzy Graphs and Interconnected Networks

Theory of the Cayley graphs is directly linked with the group theory.However,if there are uncertainties on the vertices or edges or both then fuzzy graphs have an extraordinary importance.In this perspective,numbers of generalηizations of fuzzy graphs have been explored in the literature.Among the others,picture fuzzy graph(PFG)has its own importance.A picture fuzzy graph(PFG)is a pair G=(C,D)defined on a H^(*)=(A,B),where C=(ηC,θ_(C),■_(C))is a picture fuzzy set on A and D=(ηD,θ_(D),■_(D))is a picture fuzzy set over the set B∈A×A such that for any edge mn∈ B with ηD(m,n)≤min(ηC(m),ηC(n)),θD(m,n)≤min(θC(m),θC(n))and ■_(D)(m,n)≥max(■_(C)(m),■_(C)(n)).In this manuscript,we introduce the notion of the Cayley picture fuzzy graphs on groups which is the generalization of the picture fuzzy graphs.Firstly,we discuss few important characteristics of the Cayley picture fuzzy graphs.We show that Cayley picture fuzzy graphs are vertex transitive and hence regular.Then,we investigate different types of Cayley graphs induced by the Cayley picture fuzzy graphs by using different types of cuts.We extensively discuss the term connectivity of the Cayley picture fuzzy graphs.Vertex connectivity and edge connectivity of the Cayley picture fuzzy graphs are also addressed.We also investigate the linkage between these two.Throughout,we provide the extensions of some characηteristics of both the PFGs and Cayley fuzzy graphs in the setting of Cayley picture fuzzy graphs.Finally,we provide the model of interconnected networks based on the Cayley picture fuzzy graphs.

A Note on the Inverse Connected p-Median Problem on Block Graphs

Recently, the inverse connected p-median problem on block graphs G(V,E,w) under various cost functions, say rectilinear norm, Chebyshev norm, and bottleneck Hamming distance. Their contributions include finding a necessary and sufficient condition for the connected p-median problem on block graphs, developing algorithms and showing that these problems can be solved in O(n log n) time, where n is the number of vertices in the underlying block graph. Using similar technique, we show that some results are incorrect by a counter-example. Then we redefine some notations, reprove Theorem 1 and redescribe Theorem 2, Theorem 3 and Theorem 4.

Depth First:Optimal Path Discovery Between Designated Nodes in Random Ring-Based Graphs

This paper focuses on optimally determining the existence of connected paths between some given nodes in random ring-based graphs.Serving as a fundamental underlying structure in network modeling,ring topology appears as commonplace in many realistic scenarios.Regarding this,we consider graphs composed of rings,with some possible connected paths between them.Without prior knowledge of the exact node permutations on rings,the existence of each edge can be unraveled through edge testing at a unit cost in one step.The problem examined is that of determining whether the given nodes are connected by a path or separated by a cut,with the minimum expected costs involved.Dividing the problem into different cases based on different topologies of the ring-based networks,we propose the corresponding policies that aim to quickly seek the paths between nodes.A common feature shared by all those policies is that we stick to going in the same direction during edge searching,with edge testing in each step only involving the test between the source and the node that has been tested most.The simple searching rule,interestingly,can be interpreted as a delightful property stemming from the neat structure of ring-based networks,which makes the searching process not rely on any sophisticated behaviors.We prove the optimality of the proposed policies by calculating the expected cost incurred and making a comparison with the other class of strategies.The effectiveness of the proposed policies is also verified through extensive simulations,from which we even disclose three extra intriguing findings:i)in a onering network,the cost will grow drastically with the number of designated nodes when the number is small and will grow slightly when that number is large;ii)in ring-based network,Depth First is optimal in detecting the connectivity between designated nodes;iii)the problem of multi-ring networks shares large similarity with that of two-ring networks,and a larger number of ties between rings will not influence the expected cost.

Dynamic interwell connectivity analysis of multi-layer waterflooding reservoirs based on an improved graph neural network

The analysis of interwell connectivity plays an important role in the formulation of oilfield development plans and the description of residual oil distribution. In fact, sandstone reservoirs in China's onshore oilfields generally have the characteristics of thin and many layers, so multi-layer joint production is usually adopted. It remains a challenge to ensure the accuracy of splitting and dynamic connectivity in each layer of the injection-production wells with limited field data. The three-dimensional well pattern of multi-layer reservoir and the relationship between injection-production wells can be equivalent to a directional heterogeneous graph. In this paper, an improved graph neural network is proposed to construct an interacting process mimics the real interwell flow regularity. In detail, this method is used to split injection and production rates by combining permeability, porosity and effective thickness, and to invert the dynamic connectivity in each layer of the injection-production wells by attention mechanism.Based on the material balance and physical information, the overall connectivity from the injection wells,through the water injection layers to the production layers and the output of final production wells is established. Meanwhile, the change of well pattern caused by perforation, plugging and switching of wells at different times is achieved by updated graph structure in spatial and temporal ways. The effectiveness of the method is verified by a combination of reservoir numerical simulation examples and field example. The method corresponds to the actual situation of the reservoir, has wide adaptability and low cost, has good practical value, and provides a reference for adjusting the injection-production relationship of the reservoir and the development of the remaining oil.

Restricted Edge Connectivity of Binary Undirected Kautz Graphs

A restricted edge cut is an edge cut of a connected graph whose removal resultsin a disconnected graph without isolated vertices. The size of a minimum restricted edge cutof a graph G is called its restricted edge connectivity, and is denoted by λ′(G). Let ξ(G) bethe minimum edge degree of graph G. It is known that λ′(G) ≤ξ(G) if G contains restrictededge cuts. Graph G is called maximal restricted edge connected if the equality holds in thethe preceding inequality. In this paper, undirected Kautz graph UK(2, n) is proved to bemaximal restricted edge connected if n ≥ 2.

A result on quasi k-connected graphs

Let G be a k-connected graph, and T be a subset of V（G）. If G-T is not connected,then T is said to be a cut-set of G. A k-cut-set T of G is a cut-set of G with │T│=k. Let T bea k-cut-set of a k-connected graph G. If G - T can be partitioned into subgraphs G1 and G2such that │G1│≥ 2, │G2│ 〉 2, then we call T a nontrivial k-cut-set of G. Suppose that G is a（k-1）-connected graph without nontrivial （k - 1）-cut-set. Then we call G a quasi k-connectedgraph. In this paper, we prove that for any integer k ≥ 5, if G is a k-connected graph withoutK4-, then every vertex of G is incident with an edge whose contraction yields a quasi k-connectedgraph, and so there are at least │V（G）│/2 edges of G such that the contraction of every member ofthem results in a quasi k-connected graph.

On the numbers of connected components of hattorigraph

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Brain Functional Network Generation Using Distribution-Regularized Adversarial Graph Autoencoder with Transformer for Dementia Diagnosis

The topological connectivity information derived from the brain functional network can bring new insights for diagnosing and analyzing dementia disorders.The brain functional network is suitable to bridge the correlation between abnormal connectivities and dementia disorders.However,it is challenging to access considerable amounts of brain functional network data,which hinders the widespread application of data-driven models in dementia diagnosis.In this study,a novel distribution-regularized adversarial graph auto-Encoder(DAGAE)with transformer is proposed to generate new fake brain functional networks to augment the brain functional network dataset,improving the dementia diagnosis accuracy of data-driven models.Specifically,the label distribution is estimated to regularize the latent space learned by the graph encoder,which canmake the learning process stable and the learned representation robust.Also,the transformer generator is devised to map the node representations into node-to-node connections by exploring the long-term dependence of highly-correlated distant brain regions.The typical topological properties and discriminative features can be preserved entirely.Furthermore,the generated brain functional networks improve the prediction performance using different classifiers,which can be applied to analyze other cognitive diseases.Attempts on the Alzheimer’s Disease Neuroimaging Initiative(ADNI)dataset demonstrate that the proposed model can generate good brain functional networks.The classification results show adding generated data can achieve the best accuracy value of 85.33%,sensitivity value of 84.00%,specificity value of 86.67%.The proposed model also achieves superior performance compared with other related augmentedmodels.Overall,the proposedmodel effectively improves cognitive disease diagnosis by generating diverse brain functional networks.

Average Convergence for Directed&Undirected Graphs in Distributed Systems

Consensus control of multi-agent systems is an innovative paradigm for the development of intelligent distributed systems.This has fascinated numerous scientific groups for their promising applications as they have the freedom to achieve their local and global goals and make their own decisions.Network communication topologies based on graph and matrix theory are widely used in a various real-time applications ranging from software agents to robotics.Therefore,while sustaining the significance of both directed and undirected graphs,this research emphases on the demonstration of a distributed average consensus algorithm.It uses the harmonic mean in the domain of multi-agent systems with directed and undirected graphs under static topologies based on a control input scheme.The proposed agreement protocol focuses on achieving a constant consensus on directional and undirected graphs using the exchange of information between neighbors to update their status values and to be able to calculate the total number of agents that contribute to the communication network at the same time.The proposed method is implemented for the identical networks that are considered under the directional and non-directional communication links.Two different scenarios are simulated and it is concluded that the undirected approach has an advantage over directed graph communication in terms of processing time and the total number of iterations required to achieve convergence.The same network parameters are introduced for both orientations of the communication graphs.In addition,the results of the simulation and the calculation of various matrices are provided at the end to validate the effectiveness of the proposed algorithm to achieve consensus.

A Sufficient Condition of Hamilton Connected Graph

Let G be a 3-connected graph with n vertices. The paper proves that if for each pair of vertices u and v of G, d(u,v)=2, has ｜N(u)∩N(v)｜≤α(α is the minimum independent set number), and then max{d(u),d(v)}≥n+12, then G is a Hamilton connected graph.

Note on Faudree-Schelp Theorem in Path Connected Graphs and Ore Theorem in Hamilton Connected Graphs

In this note more short proofs are given for Faudree-Schelp theorem and Ore theorem.

On Graphs with Equal Connected Domination and 2-connected Domination Numbers

A subset S of V is called a k-connected dominating set if S is a dominating set and the induced subgraph S has at most k components.The k-connected domination number γck（G） of G is the minimum cardinality taken over all minimal k-connected dominating sets of G.In this paper,we characterize trees and unicyclic graphs with equal connected domination and 2-connected domination numbers.

The Wiener Index of an Undirected Power Graph

The undirected power graph P(Zn) of a finite group Zn is the graph with vertex set G and two distinct vertices u and v are adjacent if and only if u ≠ v and or . The Wiener index W(P(Zn)) of an undirected power graph P(Zn) is defined to be sum of distances between all unordered pair of vertices in P(Zn). Similarly, the edge-Wiener index We(P(Zn)) of P(Zn) is defined to be the sum of distances between all unordered pairs of edges in P(Zn). In this paper, we concentrate on the wiener index of a power graph , P(Zpq) and P(Zp). Firstly, we obtain new results on the wiener index and edge-wiener index of power graph P(Zn), using m,n and Euler function. Also, we obtain an equivalence between the edge-wiener index and wiener index of a power graph of Zn.

An Alternative Proof of the Largest Number of Maximal Independent Sets in Connected Graphs Having at Most Two Cycles

G. C. Ying, Y. Y. Meng, B. E. Sagan, and V. R. Vatter [1] found the maximum number of maximal independent sets in connected graphs which contain at most two cycles. In this paper, we give an alternative proof to determine the largest number of maximal independent sets among all connected graphs of order n ≥ 12, which contain at most two cycles. We also characterize the extremal graph achieving this maximum value.

A TIGHT LOWER BOUND ON THE MAXIMUM GENUS OF A 3-CONNECTED LOOPLESS MULTIGRAPH

It is proved that every 3 connected loopless multigraph has maximum genus at least one third of its cycle rank plus one if its cycle rank is not less than ten, and if its cycle rank is less than ten,it is upper embeddable.This lower bound is tight.There are infinitely many 3 connected loopless multigraphs attaining this bound.

Super Cyclically Edge Connected Half Vertex Transitive Graphs

Tian and Meng in [Y. Tian and J. Meng, λc -Optimally half vertex transitive graphs with regularity k, Information Processing Letters 109 (2009) 683 - 686] shown that a connected half vertex transitive graph with regularity k and girth g(G) ≥ 6 is cyclically optimal. In this paper, we show that a connected half vertex transitive graph G is super cyclically edge-connected if minimum degree δ(G) ≥ 6 and girth g(G) ≥ 6.

Regional differences in gully network connectivity based on graph theory:a case study on the Loess Plateau,China

Material exchange frequently occurs in gullies,and thus the relationship between a gullynetwork structure and sediment transport potential has attracted considerable interest.However,previous researches ignored the difficulty of material transport from sources to sinks,and did not quantify the connectivity of a network structure.In this study,we used a graph model structure to model gully networks of six typical sample areas in the Loess Plateau of China and quantified gully network connectivity using four indexes:average node strength,accessibility from sources to sinks,potential flow,and network structural connectivity index.Results show that:(1)Reflected by different quantitative indexes,the trends of gully network connectivity in different regions are similar.From north to south,the connectivity of a sample area first increases and then decreases.(2)The more mature gullies have stronger network connectivity.Small resistance is conducive to material transport in the gullies.(3)The node connectivity index of the gully network shows a significant aggregation distribution in space,and node connectivity on the main channel is often stronger than that on the branch trench.These results not only deepen the understanding of the process and mechanism of loess gully geomorphic development and evolution but also provide a reference for geomorphic studies.

A Graph Theory Based Self-Learning Honeypot to Detect Persistent Threats

Attacks on the cyber space is getting exponential in recent times.Illegal penetrations and breaches are real threats to the individuals and organizations.Conventional security systems are good enough to detect the known threats but when it comes to Advanced Persistent Threats(APTs)they fails.These APTs are targeted,more sophisticated and very persistent and incorporates lot of evasive techniques to bypass the existing defenses.Hence,there is a need for an effective defense system that can achieve a complete reliance of security.To address the above-mentioned issues,this paper proposes a novel honeypot system that tracks the anonymous behavior of the APT threats.The key idea of honeypot leverages the concepts of graph theory to detect such targeted attacks.The proposed honey-pot is self-realizing,strategic assisted which withholds the APTs actionable tech-niques and observes the behavior for analysis and modelling.The proposed graph theory based self learning honeypot using the resultsγ(C(n,1)),γc(C(n,1)),γsc(C(n,1))outperforms traditional techniques by detecting APTs behavioral with detection rate of 96%.

结合句法增强与图注意力网络的方面级情感分类

方面级情感分类旨在识别给定特定方面文本的情感极性,在本领域中,将图神经网络与句法依赖解析相结合是当下热门的研究方向之一,此类方法通过句法解析捕捉句子中词与词之间的关系,依此构建图结构,输入图神经网络中得到情感极性。若句法解析器出现解析错误,将会对以图为基础的图神经网络模型产生巨大影响。为了增强解析器生成的句法依赖树的解析结果,文中提出了一种句法增强图注意力网络,该网络通过融合多个解析器的解析结果,提高句法依赖解析精度,得到更精准的依赖关系句法图;在图注意力网络中使用密集连接机制捕获更丰富的特征,更适配于增强后的句法图,同时引入方面注意力机制捕获方面语义特征。实验结果验证了句法增强方法的有效性,在3个基准数据集上的分类准确度都有所提高,在方面级情感分析领域具有较好的表现。

基于无向加权图上信号采样重构的推荐系统预测

为有效地捕获数据的潜在结构并且降低计算量,提出一种基于无向加权图上信号采样重构的推荐系统预测算法。为了利用未标记条目所携带的信息,将用户或项目及其关系建模为一个加权无向图。为了实现采样信号重构,在再生核希尔伯特空间中,将该问题近似地建模为一个二次无条件优化问题。为了降低计算复杂度,引入一种近似求解策略。在两个开放的公共数据库上的实验结果表明,该模型显著提高了预测精度,并且大大降低了计算复杂度。