Chain Graphs Research Articles

On the eccentricity inertia indices of chain graphs

On the eccentricity inertia indices of chain graphs

A Study on Odd Prime Labeling of Octopus Graphs Families

An odd prime labeling of a graph G (V,E), is defined as a bijective function f mapping the vertex set V to the set {1,3,5,…,2|V(G)|1}, such that for every edge uvE. the greatest common divisor gcd(f(u),f(v))=1. A graph that permits such a labeling is referred to as an odd prime graph. In this study, we explore the odd prime labeling properties of various graph structures, including the octopus chain graph, octopus ladder graph, twisted octopus ladder graph, and hexa-octopus chain graph.

Spectral deviations of graphs

Abstract For the graphs G G and H H , the spectral deviation of H H from G G is defined as ϱ G ( H ) = ∑ μ ∈ H min λ ∈ G ∣ λ − μ ∣ , {\varrho }_{G}\left(H)=\sum _{\mu \in H}\mathop{\min }\limits_{\lambda \in G}| \lambda -\mu | , where ∈ \in designates that the given number is an eigenvalue of the adjacency matrix of the corresponding graph. In this study, we consider the problem of existence of a proper induced subgraph H H of a prescribed graph G G such that ϱ G ( H ) = 0 {\varrho }_{G}\left(H)=0 , and the problem of determination of all such subgraphs. We investigate these problems in the framework of Smith graphs and their induced subgraphs, graphs with small second largest eigenvalue, graphs with small number of either positive or distinct eigenvalues, integral graphs, and chain graphs. Our results can be interesting in the context of graphs with a fixed number of distinct eigenvalues, eigenvalue distribution, or spectral distances of graphs.

A Novel Approach to Cexp Average Assignments on Chain Graphs

بشكل عام، ليس من الضروري أن يكون المتوسط ​​الأسي لعددين موجبين عددًا صحيحًا. ولهذا السبب، يجب أن يكون المتوسط ​​الأسي عددًا صحيحًا يأخذ في الاعتبار دالة الأرضية أو السقف. لقد تم تعريفها بحيث يمكن تسمية الرسوم البيانية بمتوسط ​​أسي، حيث يمكن لدالة الأرضية أو دالة السقف تطبيقها على تسميات الحواف. لتأسيس تعيين المتوسط ​​الأسي على الرسوم البيانية، سوف يتم وضع تسميات الحواف التي تنشأ من دالة السقف وحدها في الاعتبار. تُسمى دالة تعيين قمة الرأس δ ودالة تعيين الحافة بتخصيص متوسط ​​Cexp للرسم البياني G مع رؤوس p وحواف q إذا كانت δ شاملة و متباينة وتكون العلاقات المكافئة ويتم تعريفه بواسطة تسمية الحافة δ* كما يلي: , حيث وN هي مجموعة جميع الأعداد الطبيعية. إذا كان الرسم البياني يقبل تعيين متوسط ​​Cexp، فإنه يطلق عليه رسم بياني لمتوسط ​​تخصيصCexp. يُقترح في هذه الورقة متوسط ​​تخصيص الرسوم البيانية لـCexp ، ويتم استكشاف خصائصه في الدورة، واتحاد المسار والدورة، واتحاد الرسم البياني T والدورة، والرسم البياني G*، والرسم البياني G'، والرسم البياني Ĝ والشرغوف.

Risk factors for fluoroquinolone- and macrolide-resistance among swine Campylobacter coli using multi-layered chain graphs.

Campylobacter spp. resistant to fluoroquinolones and macrolides are serious public health threats. Studies aiming to identify risk factors for drug-resistant Campylobacter have narrowly focused on antimicrobial use at the farm level. Using chain graphs, we quantified risk factors for fluoroquinolones- and macrolide-resistance in Campylobacter coli isolated from two distinctive swine production systems, conventional and antibiotic-free (ABF). The chain graphs were learned using genotypic and phenotypic resistance data from 1082 isolates and host exposures obtained through surveys for 18 cohorts of pigs. The gyrA T86I point mutation alone explained at least 58 % of the variance in ciprofloxacin minimum inhibitory concentration (MIC) for ABF and 79 % in conventional farms. For macrolides, genotype and host exposures explained similar variance in azithromycin and erythromycin MIC. Among host exposures, heavy metal exposures were identified as risk factors in both conventional and ABF. Chain graph models can generate insights into the complex epidemiology of antimicrobial resistance by characterizing context-specific risk factors and facilitating causal discovery.

THE LOCATING CHROMATIC NUMBER OF CHAIN(A,4,n) GRAPH

Let be a connected graph with a vertex coloringsuch that two adjacent vertices have different colors. We denote an ordered partition where is a color class with color-, consisting of vertices given color , for . The color code of a vertex in is a -vector: . where is the distance between a vertex in and for . If every two vertices and in have different color codes, , then is called the locating -coloring of . The minimum number of colors k needed in this coloring is defined as the locating chromatic number, denoted by . This paper determines the locating chromatic number of chain graph and the induction of two graphs . Graph is a cyclic graph , which is the identification of , for n>2.

Bayesian robust learning in chain graph models for integrative pharmacogenomics

Integrative analysis of multi-level pharmacogenomic data for modeling dependencies across various biological domains is crucial for developing genomic-testing based treatments. Chain graphs characterize conditional dependence structures of such multi-level data where variables are naturally partitioned into multiple ordered layers, consisting of both directed and undirected edges. Existing literature mostly focus on Gaussian chain graphs, which are ill-suited for non-normal distributions with heavy-tailed marginals, potentially leading to inaccurate inferences. We propose a Bayesian robust chain graph model (RCGM) based on random transformations of marginals using Gaussian scale mixtures to account for node-level non-normality in continuous multivariate data. This flexible modeling strategy facilitates identification of conditional sign dependencies among non-normal nodes while still being able to infer conditional dependencies among normal nodes. In simulations, we demonstrate that RCGM outperforms existing Gaussian chain graph inference methods in data generated from various non-normal mechanisms. We apply our method to genomic, transcriptomic and proteomic data to understand underlying biological processes holistically for drug response and resistance in lung cancer cell lines. Our analysis reveals inter-and intra-platform dependencies of key signaling pathways to monotherapies of icotinib, erlotinib and osimertinib among other drugs, along with shared patterns of molecular mechanisms behind drug actions.

Typhoon related cascading fault chain dynamic evolution model and risk mitigation in distribution systems

Typhoon related cascading fault chain dynamic evolution model and risk mitigation in distribution systems

On the eccentricity spectra of chain graphs

On the eccentricity spectra of chain graphs

The number of spanning trees in K n -chain and ring graphs

The problem of counting spanning trees of graphs or networks is a fundamental and crucial area of research in combinatorics, while has numerous important applications in statistical physics, network theory and theoretical computer science. Very recently, Kosar, Zaman, Ali and Ullah obtained a nice formula on the number of spanning trees of a K 5-chain network K5ℓ constructed by connecting ℓcopys of complete graphs K 5. They made extensive use of matrix theory and spectral graph theory, especially the normalized Laplacian of graphs. In this paper, by using a rather simple and more physical treatment (the mesh-star transformation in electrical network) without any linear algebra, we generalize their result to K n -chain graphs and K n -ring graphs. The results show that there is a simple relation between the number of spanning trees of the K n -chain graph Lnℓ and the K n -ring graph Cnℓ . We also calculate the corresponding tree entropy (or so called ‘the asymptotic growth constant’) and find that the tree entropy of the corresponding K n -chain graphs and K n -ring graphs are totally the same.

On Sombor index and graph energy of some chemically important graphs

Sombor index of a graph G=(V(G),E(G)) is provided by the expression ∑uv∈E(G)du2+dv2, where dx is the degree of the vertex x∈V(G). The energy of a graph is the quantity given by the total of the absolute values of its adjacency matrix’s eigenvalues. In this article, we improve the relation between the Sombor index and graph energy and derive the relation between them for unicyclic, bicyclic and tricyclic graphs, trees, triangular chain, square cactus chain and hexagonal cactus chain graphs. At last, we find the bounds of graph energy for zigzag and linear hexagonal chains.

Canonical cuts of path powers

The MaxCut problem aims to find a bipartition of vertices in a given graph such that the number of edges with one end vertex in each part is maximum among all bipartitions. NP-hardness when restricted to interval graphs has been recently announced. Surprisingly, all previously published attempts at polynomial-time algorithms for unit interval graphs turned out to be wrong, which justifies the search for subclasses where MaxCut can be handled. We introduce canonical cuts whose pattern allows an easy computation of the cut size for the power of paths $P_n^k$. Using canonical cuts, we calculate the structure and the size of maximum cuts for $k\leq 5$ and for $n\leq \frac{2}{3}(2k+2)$. We prove that the known size for a maximum cut for reduced co-bipartite chain graphs can be achieved by a canonical cut. We perform computational experiments on each $P_n^k$ graph with $1\leq k\leq n\leq 43$ and show that most of them allow a canonical cut that is maximum. We display a table with the found cases where there is no canonical cut which is a maximum cut. In these graphs, the difference between the maximum cut and some canonical is at most 3 units. This indicates canonical cuts as a good approach to tackle the maximum cut on $P_n^k$ graphs.

Chain graph structure learning based on minimal c-separation trees

Chain graph structure learning based on minimal c-separation trees

Some Algorithmic Results for Eternal Vertex Cover Problem in Graphs

The eternal vertex cover problem is a variant of the vertex cover problem. It is a two-player (attacker and defender) game in which, given a graph $G=(V,E)$, the defender needs to allocate guards at some vertices so that the allocated vertices form a vertex cover. The attacker can attack one edge at a time, and the defender needs to move the guards along the edges such that at least one guard moves through the attacked edge and the new configuration still remains a vertex cover. The attacker wins if no such move exists for the defender. The defender wins if there exists a strategy to defend the graph against infinite sequence of attacks. The minimum number of guards with which the defender can form a winning strategy is called the eternal vertex cover number of $G$, and is denoted by $evc(G)$. Given a graph $G$, the problem of finding the eternal vertex cover number is NP-hard for general graphs and remains NP-hard even for bipartite graphs. We have given a polynomial time algorithm to find the Eternal vertex cover number in chain graphs and $P_4$-sparse graphs. We have also given a linear-time algorithm to find the eternal vertex cover number for split graphs, an important subclass of chordal graphs.

Certifying Induced Subgraphs in Large Graphs

We introduce I/O-efficient certifying algorithms for the recognition of bipartite, split, threshold, bipartite chain, and trivially perfect graphs. When the input graph is a member of the respective class, the certifying algorithm returns a certificate that characterizes this class.Otherwise, it returns a forbidden induced subgraph as a certificate for non-membership.On a graph with $n$ vertices and $m$ edges, our algorithms take $\mathcal O(\text{sort}(n + m))$ I/Os in the worst case for split, threshold and trivially perfect graphs.In the same complexity bipartite and bipartite chain graphs can be certified with high probability.We provide implementations and an experimental evaluation for split and threshold graphs.

A Semantic Knowledge Graph of European Mountain Value Chains

The United Nations forecast a significant shift in global population distribution by 2050, with rural populations projected to decline. This decline will particularly challenge mountain areas’ cultural heritage, well-being, and economic sustainability. Understanding the economic, environmental, and societal effects of rural population decline is particularly important in Europe, where mountainous regions are vital for supplying goods. The present paper describes a geospatially explicit semantic knowledge graph containing information on 454 European mountain value chains. It is the first large-size, structured collection of information on mountain value chains. Our graph, structured through ontology-based semantic modelling, offers representations of the value chains in the form of narratives. The graph was constructed semi-automatically from unstructured data provided by mountain-area expert scholars. It is accessible through a public repository and explorable through interactive Story Maps and a semantic Web service. Through semantic queries, we demonstrate that the graph allows for exploring territorial complexities and discovering new knowledge on mountain areas’ environmental, societal, territory, and economic aspects that could help stem depopulation.

Distributed Partial Quantum Consensus of Qubit Networks With Connected Topologies.

In this article, we consider the partial quantum consensus problem of a qubit network in a distributed view. The local quantum operation is designed based on the Hamiltonian by using the local information of each quantum system in a network of qubits. We construct the unitary transformation for each quantum system to achieve the partial quantum consensus, that is, the directions of the quantum states in the Bloch ball will reach an agreement. A simple case of two-qubit quantum systems is considered first, and a minimum completing time of reaching partial consensus is obtained based on the geometric configuration of each qubit. Furthermore, we extend the approaches to deal with the more general N -qubit networks. Two partial quantum consensus protocols, based on the Lyapunov method for chain graphs and the geometry method for connected graphs, are proposed. The geometry method can be utilized to deal with more general connected graphs, while for the Lyapunov method, the global consensus can be obtained. The numerical simulation over a qubit network is demonstrated to verify the validity and the effectiveness of the theoretical results.

Reasoning graph-based reinforcement learning to cooperate mixed connected and autonomous traffic at unsignalized intersections

Reasoning graph-based reinforcement learning to cooperate mixed connected and autonomous traffic at unsignalized intersections

Learning the structure of multivariate regression chain graphs by testing complete separators in prime blocks

This paper introduces an algorithm to construct a bidirectional causal graph using an augmented graph. The algorithm decomposes the augmented graph, significantly reducing the size of the variable set required for conditional independence testing. Simultaneously, it preserves the fundamental structure of the augmented graph after decomposition, saving time and cost in constructing a global skeleton graph. Through experiments on discrete and continuous datasets, the algorithm demonstrates a clear advantage in runtime compared to traditional methods. In large-scale sparse networks, the training time is only about one-tenth of traditional methods. Additionally, the algorithm shows improvement in terms of construction error. Since the input to the algorithm is an augmented graph, this paper also discusses the impact on construction error when using both real and generated augmented graphs as input. Furthermore, the concept of markov blanket is extended to multivariate regression chain graphs, providing a method for rapidly constructing augmented graphs given certain prior knowledge.

Handling Efficient VNF Placement with Graph-Based Reinforcement Learning for SFC Fault Tolerance

Network functions virtualization (NFV) has become the platform for decomposing the sequence of virtual network functions (VNFs), which can be grouped as a forwarding graph of service function chaining (SFC) to serve multi-service slice requirements. NFV-enabled SFC consists of several challenges in reaching the reliability and efficiency of key performance indicators (KPIs) in management and orchestration (MANO) decision-making control. The problem of SFC fault tolerance is one of the most critical challenges for provisioning service requests, and it needs resource availability. In this article, we proposed graph neural network (GNN)-based deep reinforcement learning (DRL) to enhance SFC fault tolerance (GRL-SFT), which targets the chain graph representation, long-term approximation, and self-organizing service orchestration for future massive Internet of Everything applications. We formulate the problem as the Markov decision process (MDP). DRL seeks to maximize the cumulative rewards by maximizing the service request acceptance ratios and minimizing the average completion delays. The proposed model solves the VNF management problem in a short time and configures the node allocation reliably for real-time restoration. Our simulation result demonstrates the effectiveness of the proposed scheme and indicates better performance in terms of total rewards, delays, acceptances, failures, and restoration ratios in different network topologies compared to reference schemes.