Search results for: graph analysis

Authors: David Pham, Yongfeng Zhang

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Graphs are an important structure for data storage and computation. Recent years have seen the success of deep learning on graphs such as Graph Neural Networks (GNN) on various data mining and machine learning tasks. However, most of the deep learning models on graphs cannot easily explain their predictions and are thus often labelled as “black boxes.” For example, Graph Attention Network (GAT) is a frequently used GNN architecture, which adopts an attention mechanism to carefully select the neighborhood nodes for message passing and aggregation. However, it is difficult to explain why certain neighbors are selected while others are not and how the selected neighbors contribute to the final classification result. In this paper, we present a graph learning model called Explainable Graph Attention Network (XGAT), which integrates graph attention modeling and explainability. We use a single model to target both the accuracy and explainability of problem spaces and show that in the context of graph attention modeling, we can design a unified neighborhood selection strategy that selects appropriate neighbor nodes for both better accuracy and enhanced explainability. To justify this, we conduct extensive experiments to better understand the behavior of our model under different conditions and show an increase in both accuracy and explainability.

Keywords: explainable AI, graph attention network, graph neural network, node classification

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Authors: Zainab Yasir Al-Rekaby, Abdul Jalil M. Khalaf

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Coloring the vertices of a graph such that every two adjacent vertices have different color is a very common problem in the graph theory. This is known as proper coloring of graphs. The possible number of different proper colorings on a graph with a given number of colors can be represented by a function called the chromatic polynomial. Two graphs G and H are said to be chromatically equivalent, if they share the same chromatic polynomial. A Graph G is chromatically unique, if G is isomorphic to H for any graph H such that G is chromatically equivalent to H. The study of chromatically equivalent and chromatically unique problems is called chromaticity. This paper shows that a wheel W14 is chromatically unique.

Keywords: chromatic polynomial, chromatically Equivalent, chromatically unique, wheel

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Authors: Xiaoheng Tan, Xianfang Li, Tan Guo, Yuchuan Liu, Zhijun Yang, Hongye Li, Kai Fu, Yufang Wu, Heling Gong

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The image data collected in reality often have high dimensions, and it contains noise and redundant information. Therefore, it is necessary to extract the compact feature expression of the original perceived image. In this process, effective use of prior knowledge such as data structure distribution and sample label is the key to enhance image feature discrimination and robustness. Based on the above considerations, this paper proposes a local preserving discriminant feature learning model based on graph optimization. The model has the following characteristics: (1) Locality preserving constraint can effectively excavate and preserve the local structural relationship between data. (2) The flexibility of graph learning can be improved by constructing a new local geometric structure graph using label information and the nearest neighbor threshold. (3) The L₂,₁ norm is used to redefine LDA, and the diagonal matrix is introduced as the scale factor of LDA, and the samples are selected, which improves the robustness of feature learning. The validity and robustness of the proposed algorithm are verified by experiments in two public image datasets.

Keywords: feature extraction, graph optimization local preserving projection, linear discriminant analysis, L₂, ₁ norm

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Authors: Po-Fang Hsu, Chiching Wei

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In this paper, we present a novel neural graph matching approach applied to document comparison. Document comparison is a common task in the legal and financial industries. In some cases, the most important differences may be the addition or omission of words, sentences, clauses, or paragraphs. However, it is a challenging task without recording or tracing the whole edited process. Under many temporal uncertainties, we explore the potentiality of our approach to proximate the accurate comparison to make sure which element blocks have a relation of edition with others. In the beginning, we apply a document layout analysis that combines traditional and modern technics to segment layouts in blocks of various types appropriately. Then we transform this issue into a problem of layout graph matching with textual awareness. Regarding graph matching, it is a long-studied problem with a broad range of applications. However, different from previous works focusing on visual images or structural layout, we also bring textual features into our model for adapting this domain. Specifically, based on the electronic document, we introduce an encoder to deal with the visual presentation decoding from PDF. Additionally, because the modifications can cause the inconsistency of document layout analysis between modified documents and the blocks can be merged and split, Sinkhorn divergence is adopted in our neural graph approach, which tries to overcome both these issues with many-to-many block matching. We demonstrate this on two categories of layouts, as follows., legal agreement and scientific articles, collected from our real-case datasets.

Keywords: document comparison, graph matching, graph neural network, modification similarity, multi-modal

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Authors: Jiajun Wang, Xiaoge Li

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The purpose of the aspect-level sentiment analysis task is to identify the sentiment polarity of aspects in a sentence. Currently, most methods mainly focus on using neural networks and attention mechanisms to model the relationship between aspects and context, but they ignore the dependence of words in different ranges in the sentence, resulting in deviation when assigning relationship weight to other words other than aspect words. To solve these problems, we propose a new aspect-level sentiment analysis model that combines a multi-channel convolutional network and graph convolutional network (GCN). Firstly, the context and the degree of association between words are characterized by Long Short-Term Memory (LSTM) and self-attention mechanism. Besides, a multi-channel convolutional network is used to extract the features of words in different ranges. Finally, a convolutional graph network is used to associate the node information of the dependency tree structure. We conduct experiments on four benchmark datasets. The experimental results are compared with those of other models, which shows that our model is better and more effective.

Keywords: aspect-level sentiment analysis, attention, multi-channel convolution network, graph convolution network, dependency tree

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Authors: Junjie Zhong, Kai Hong, Lei Wang

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Passage Retrieval (PR) plays an important role in many Natural Language Processing (NLP) tasks. Traditional efficient retrieval models relying on exact term-matching, such as TF-IDF or BM25, have nowadays been exceeded by pre-trained language models which match by semantics. Though they gain effectiveness, deep language models often require large memory as well as time cost. To tackle the trade-off between efficiency and effectiveness in PR, this paper proposes Graph Passage Retriever (GraphPR), a graph-based model inspired by the development of graph learning techniques. Different from existing works, GraphPR is end-to-end and integrates both term-matching information and semantics. GraphPR constructs a passage-level graph from BM25 retrieval results and trains a GCN-like model on the graph with graph-based objectives. Passages were regarded as nodes in the constructed graph and were embedded in dense vectors. PR can then be implemented using embeddings and a fast vector-similarity search. Experiments on a variety of real-world retrieval datasets show that the proposed model outperforms related models in several evaluation metrics (e.g., mean reciprocal rank, accuracy, F1-scores) while maintaining a relatively low query latency and memory usage.

Keywords: efficiency, effectiveness, graph learning, language model, passage retrieval, term-matching model

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Authors: Soner Nandappa D., Ahmed Mohammed Naji

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In a graph G = (V, E), the distance from a vertex v to a vertex u is the length of shortest v to u path. The eccentricity e(v) of v is the distance to a farthest vertex from v. The diameter diam(G) is the maximum eccentricity. The k-distance neighborhood of v, for 0 ≤ k ≤ e(v), is Nk(v) = {u ϵ V (G) : d(v, u) = k}. In this paper, we introduce a new distance degree based topological polynomial of a graph G is called a k- distance neighborhood polynomial, denoted Nk(G, x). It is a polynomial with the coefficient of the term k, for 0 ≤ k ≤ e(v), is the sum of the cardinalities of Nk(v) for every v ϵ V (G). Some properties of k- distance neighborhood polynomials are obtained. Exact formulas of the k- distance neighborhood polynomial for some well-known graphs, Cartesian product and join of graphs are presented.

Keywords: vertex degrees, distance in graphs, graph operation, Nk-polynomials

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Authors: Shuo Liu

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In Chinese text classification tasks, redundant words and phrases can interfere with the formation of extracted and analyzed text information, leading to a decrease in the accuracy of the classification model. To reduce irrelevant elements, extract and utilize text content information more efficiently and improve the accuracy of text classification models. In this paper, the text in the corpus is first extracted using the TextRank algorithm for abstraction, the words in the abstract are used as nodes to construct a text graph, and then the graph attention network (GAT) is used to complete the task of classifying the text. Testing on a Chinese dataset from the network, the classification accuracy was improved over the direct method of generating graph structures using text.

Keywords: Chinese natural language processing, text classification, abstract extraction, graph attention network

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Authors: Mehrshad Khosraviani, Sepehr Najjarpour

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Considering the usage of graph-based approaches in systems and synthetic biology, and the various types of ‎the graphs employed by them, a comprehensive graph library based ‎on the three-tier architecture (3TA) was previously introduced for full representation of the biological systems. Although proposing a 3TA-based graph library, three following reasons motivated us to redesign the graph ‎library based on the service-oriented architecture (SOA): (1) Maintaining the accuracy of the data related to an input graph (including its edges, its ‎vertices, its topology, etc.) without involving the end user:‎ Since, in the case of using 3TA, the library files are available to the end users, they may ‎be utilized incorrectly, and consequently, the invalid graph data will be provided to the ‎computer algorithms. However, considering the usage of the SOA, the operation of the ‎graph registration is specified as a service by encapsulation of the library files. In other words, overall control operations needed for registration of the valid data will be the ‎responsibility of the services. (2) Partitioning of the library product into some different parts: Considering 3TA, a whole library product was provided in general. While here, the product ‎can be divided into smaller ones, such as an AND/OR graph drawing service, and each ‎one can be provided individually. As a result, the end user will be able to select any ‎parts of the library product, instead of all features, to add it to a project. (3) Reduction of the complexities: While using 3TA, several other libraries must be needed to add for connecting to the ‎database, responsibility of the provision of the needed library resources in the SOA-‎based graph library is entrusted with the services by themselves. Therefore, the end user ‎who wants to use the graph library is not involved with its complexity. In the end, in order to ‎make ‎the library easier to control in the system, and to restrict the end user from accessing the files, ‎it was preferred to use the service-oriented ‎architecture ‎‎(SOA) over the three-tier architecture (3TA) and to redevelop the previously proposed graph library based on it‎.

Keywords: Bio-Design Automation, Biological System, Graph Library, Service-Oriented Architecture, Systems and Synthetic Biology

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Authors: Shaowei Sun

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Let G be a graph with vertex set V(G)={v_1,v_2,...,v_n} and edge set E(G). For any vertex v belong to V(G), let d_v denote the degree of v. The normalized Laplacian matrix of the graph G is the matrix where the non-diagonal (i,j)-th entry is -1/(d_id_j) when vertex i is adjacent to vertex j and 0 when they are not adjacent, and the diagonal (i,i)-th entry is the di. In this paper, we discuss some bounds on the largest and the second smallest normalized Laplacian eigenvalue of trees and graphs. As following, we found some new bounds on the second smallest normalized Laplacian eigenvalue of tree T in terms of graph parameters. Moreover, we use Sage to give some conjectures on the second largest and the third smallest normalized eigenvalues of graph.

Keywords: graph, normalized Laplacian eigenvalues, normalized Laplacian matrix, tree

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Authors: Alfi Y. Zakiyyah, Hanni Garminia, M. Salman, A. N. Irawati

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In the terminology of the hypergraph, there is a relation with the terminology graph. In the theory of graph, the edges connected two vertices. In otherwise, in hypergraph, the edges can connect more than two vertices. There is representation matrix of a graph such as adjacency matrix, Laplacian matrix, and incidence matrix. The adjacency matrix is symmetry matrix so that all eigenvalues is real. This matrix is a nonnegative matrix. The all diagonal entry from adjacency matrix is zero so that the trace is zero. Another representation matrix of the graph is the Laplacian matrix. Laplacian matrix is symmetry matrix and semidefinite positive so that all eigenvalues are real and non-negative. According to the spectral study in the graph, some that result is generalized to hypergraph. A hypergraph can be represented by a matrix such as adjacency, incidence, and Laplacian matrix. Throughout for this term, we use Laplacian matrix to represent a complete tripartite hypergraph. The aim from this research is to determine second smallest eigenvalues from this matrix and find a relation this eigenvalue with the connectivity of that hypergraph.

Keywords: connectivity, graph, hypergraph, Laplacian matrix

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Authors: Robert Gold

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Control flow graphs are a well-known representation of the sequential control flow structure of programs with a multitude of applications. Not only single functions but also sets of functions or complete programs can be modelled by control flow graphs. In this case the size of the graphs can grow considerably and thus makes it difficult for software engineers to analyse the control flow. Graph reductions are helpful in this situation. In this paper we define reductions to subsets of nodes. Since executions of programs are represented by paths through the control flow graphs, paths should be preserved. Furthermore, the composition of reductions makes a stepwise analysis approach possible.

Keywords: control flow graph, graph reduction, software engineering, software applications

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Authors: Shin-Shin Kao, Yuan-Kang Shih, Hsun Su

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In this paper, we show that the conjecture of Chv tal, which states that any 1-tough graph is either a Hamiltonian graph or its complement contains a specific graph denoted by F, does not hold in general. More precisely, it is true only for graphs with six or seven vertices, and is false for graphs with eight or more vertices. A theorem is derived as a correction for the conjecture.

Keywords: complement, degree sum, hamiltonian, tough

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Authors: Youhang Zhou, Weimin Zeng, Qi Xie

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Abstract: In order to solve the wear recognition problem of the machine tool guide surface, a new machine tool guide surface recognition method based on the radar-graph barycentre feature is presented in this paper. Firstly, the gray mean value, skewness, projection variance, flat degrees and kurtosis features of the guide surface image data are defined as primary characteristics. Secondly, data Visualization technology based on radar graph is used. The visual barycentre graphical feature is demonstrated based on the radar plot of multi-dimensional data. Thirdly, a classifier based on the support vector machine technology is used, the radar-graph barycentre feature and wear original feature are put into the classifier separately for classification and comparative analysis of classification and experiment results. The calculation and experimental results show that the method based on the radar-graph barycentre feature can detect the guide surface effectively.

Keywords: guide surface, wear defects, feature extraction, data visualization

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Authors: Lijuan Zhou, Mengqi Wu, Changyong Niu

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Attributed graph clustering can utilize the graph topology and node attributes to uncover hidden community structures and patterns in complex networks, aiding in the understanding and analysis of complex systems. Utilizing contrastive learning for attributed graph clustering can effectively exploit meaningful implicit relationships between data. However, existing attributed graph clustering methods based on contrastive learning suffer from the following drawbacks: 1) Complex data augmentation increases computational cost, and inappropriate data augmentation may lead to semantic drift. 2) The selection of positive and negative samples neglects the intrinsic cluster structure learned from graph topology and node attributes. Therefore, this paper proposes a method called self-supervised Attributed Graph Clustering with Dual Contrastive Loss constraints (AGC-DCL). Firstly, Siamese Multilayer Perceptron (MLP) encoders are employed to generate two views separately to avoid complex data augmentation. Secondly, the neighborhood contrastive loss is introduced to constrain node representation using local topological structure while effectively embedding attribute information through attribute reconstruction. Additionally, clustering-oriented contrastive loss is applied to fully utilize clustering information in global semantics for discriminative node representations, regarding the cluster centers from two views as negative samples to fully leverage effective clustering information from different views. Comparative clustering results with existing attributed graph clustering algorithms on six datasets demonstrate the superiority of the proposed method.

Keywords: attributed graph clustering, contrastive learning, clustering-oriented, self-supervised learning

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Authors: Amir Bahrami

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In the fields of chemical graph theory, molecular topology, and mathematical chemistry, a topological index or a descriptor index also known as a connectivity index is a type of a molecular descriptor that is calculated based on the molecular graph of a chemical compound. Topological indices are numerical parameters of a graph which characterize its topology and are usually graph invariant. Topological indices are used for example in the development of quantitative structure-activity relationships (QSARs) in which the biological activity or other properties of molecules are correlated with their chemical structure. In this paper some descriptor index (descriptor index) of single-walled carbon nanotubes, is determined.

Keywords: chemical graph theory, molecular topology, molecular descriptor, single-walled carbon nanotubes

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Authors: Bing Li, Zhi Li, Yilong Yang

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Web classification can promote the quality of service discovery and management in the service repository. It is widely used to locate developers desired services. Although traditional classification methods based on supervised learning models can achieve classification tasks, developers need to manually mark web services, and the quality of these tags may not be enough to establish an accurate classifier for service classification. With the doubling of the number of web services, the manual tagging method has become unrealistic. In recent years, the attention mechanism has made remarkable progress in the field of deep learning, and its huge potential has been fully demonstrated in various fields. This paper designs a multi-head attention graph convolutional network (MHAGCN) service classification method, which can assign different weights to the neighborhood nodes without complicated matrix operations or relying on understanding the entire graph structure. The framework combines the advantages of the attention mechanism and graph convolutional neural network. It can classify web services through automatic feature extraction. The comprehensive experimental results on a real dataset not only show the superior performance of the proposed model over the existing models but also demonstrate its potentially good interpretability for graph analysis.

Keywords: attention mechanism, graph convolutional network, interpretability, service classification, service discovery

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Authors: Ould Bouamama, M. Bressel, D. Hissel, M. Hilairet

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Fuel cell (FC) is one of the best alternatives of fossil energy. Recently, the research community of fuel cell has shown a considerable interest for diagnosis in view to ensure safety, security, and availability when faults occur in the process. The problematic for model based FC diagnosis consists in that the model is complex because of coupling of several kind of energies and the numerical values of parameters are not always known or are uncertain. The present paper deals with use of one tool: the Linear Fractional Transformation bond graph tool not only for uncertain modelling but also for monitorability (ability to detect and isolate faults) analysis and formal generation of robust fault indicators with respect to parameter uncertainties.The developed theory applied to a nonlinear FC system has proved its efficiency.

Keywords: bond graph, fuel cell, fault detection and isolation (FDI), robust diagnosis, structural analysis

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Authors: Yasmeen A. S. Essawy, Khaled Nassar

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With the rapid increase of complexity in the building industry, professionals in the A/E/C industry were forced to adopt Building Information Modeling (BIM) in order to enhance the communication between the different project stakeholders throughout the project life cycle and create a semantic object-oriented building model that can support geometric-topological analysis of building elements during design and construction. This paper presents a model that extracts topological relationships and geometrical properties of building elements from an existing fully designed BIM, and maps this information into a directed acyclic Elemental Graph Data Model (EGDM). The model incorporates BIM-based search algorithms for automatic deduction of geometrical data and topological relationships for each building element type. Using graph search algorithms, such as Depth First Search (DFS) and topological sortings, all possible construction sequences can be generated and compared against production and construction rules to generate an optimized construction sequence and its associated schedule. The model is implemented in a C# platform.

Keywords: building information modeling (BIM), elemental graph data model (EGDM), geometric and topological data models, graph theory

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Authors: Aicha Majda, Abdelhamid El Hassani

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Lung CT image segmentation is a prerequisite in lung CT image analysis. Most of the conventional methods need a post-processing to deal with the abnormal lung CT scans such as lung nodules or other lesions. The simplest similarity measure in the standard Graph Cuts Algorithm consists of directly comparing the pixel values of the two neighboring regions, which is not accurate because this kind of metrics is extremely sensitive to minor transformations such as noise or other artifacts problems. In this work, we propose an improved version of the standard graph cuts algorithm based on the Patch-Based similarity metric. The boundary penalty term in the graph cut algorithm is defined Based on Patch-Based similarity measurement instead of the simple intensity measurement in the standard method. The weights between each pixel and its neighboring pixels are Based on the obtained new term. The graph is then created using theses weights between its nodes. Finally, the segmentation is completed with the minimum cut/Max-Flow algorithm. Experimental results show that the proposed method is very accurate and efficient, and can directly provide explicit lung regions without any post-processing operations compared to the standard method.

Keywords: graph cuts, lung CT scan, lung parenchyma segmentation, patch-based similarity metric

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Authors: Shokofeh Ebrtahimi

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Let G be the chemical graph of a molecule. The matrix D = [dij ] is called the detour matrix of G, if dij is the length of longest path between atoms i and j. The sum of all entries above the main diagonal of D is called the detour index of G.Chemical graph theory is the topology branch of mathematical chemistry which applies graph theory to mathematical modelling of chemical phenomena.[1] The pioneers of the chemical graph theory are Alexandru Balaban, Ante Graovac, Ivan Gutman, Haruo Hosoya, Milan Randić and Nenad TrinajstićLet G be the chemical graph of a molecule. The matrix D = [dij ] is called the detour matrix of G, if dij is the length of longest path between atoms i and j. The sum of all entries above the main diagonal of D is called the detour index of G. In this paper, a new program for computing the detour index of molecular graphs of nanotubes by heptagons is determineded. Some Conjectures about detour index of Molecular graphs of nanotubes is included.

Keywords: chemical graph, detour matrix, Detour index, carbon nanotube

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Authors: Yingqi Cui, Changran Huang, Raymond Lee

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In this paper, we proposed an AI Tutor using ontology and natural language process techniques to generate a computer science domain knowledge graph and answer users’ questions based on the knowledge graph. We define eight types of relation to extract relationships between entities according to the computer science domain text. The AI tutor is separated into two agents: learning agent and Question-Answer (QA) agent and developed on JADE (a multi-agent system) platform. The learning agent is responsible for reading text to extract information and generate a corresponding knowledge graph by defined patterns. The QA agent can understand the users’ questions and answer humans’ questions based on the knowledge graph generated by the learning agent.

Keywords: artificial intelligence, natural Language processing, knowledge graph, intelligent agents, QA system

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Authors: Mustafa Elfituri, Jonathan Cook

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Recently, graph-based computations have become more important in large-scale scientific computing as they can provide a methodology to model many types of relations between independent objects. They are being actively used in fields as varied as biology, social networks, cybersecurity, and computer networks. At the same time, graph problems have some properties such as irregularity and poor locality that make their performance different than regular applications performance. Therefore, parallelizing graph algorithms is a hard and challenging task. Initial evidence is that standard computer architectures do not perform very well on graph algorithms. Little is known exactly what causes this. The Graph500 benchmark is a representative application for parallel graph-based computations, which have highly irregular data access and are driven more by traversing connected data than by computation. In this paper, we present results from analyzing the performance of various example implementations of Graph500, including a shared memory (OpenMP) version, a distributed (MPI) version, and a hybrid version. We measured and analyzed all the factors that affect its performance in order to identify possible changes that would improve its performance. Results are discussed in relation to what factors contribute to performance degradation.

Keywords: graph computation, graph500 benchmark, parallel architectures, parallel programming, workload characterization.

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Authors: Bin Sheng, Changhong Lu

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Let G be a simple undirected graph with no isolated vertex. A paired dominating set of G is a dominating set which induces a subgraph that has a perfect matching. The paired domination number of G, denoted by γₚᵣ(G), is the size of its smallest paired dominating set. Goddard and Henning conjectured that γₚᵣ(G) ≤ 4n/7 holds for every graph G with δ(G) ≥ 3, except the Petersen Graph. In this paper, we prove this conjecture for cubic graphs.

Keywords: paired dominating set, upper bound, cubic graphs, weight function

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Authors: Jake Gonzalez, Tommy Dang

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This paper proposes a temporal graph approach to visualize and analyze the evolution of gene relationships and nomenclature over time. An interactive web-based tool implements this temporal graph, enabling researchers to traverse a timeline and observe coupled dynamics in network topology and naming conventions. Analysis of a real human genomic dataset reveals the emergence of densely interconnected functional modules over time, representing groups of genes involved in key biological processes. For example, the antimicrobial peptide DEFA1A3 shows increased connections to related alpha-defensins involved in infection response. Tracking degree and betweenness centrality shifts over timeline iterations also quantitatively highlight the reprioritization of certain genes’ topological importance as knowledge advances. Examination of the CNR1 gene encoding the cannabinoid receptor CB1 demonstrates changing synonymous relationships and consolidating naming patterns over time, reflecting its unique functional role discovery. The integrated framework interconnecting these topological and nomenclature dynamics provides richer contextual insights compared to isolated analysis methods. Overall, this temporal graph approach enables a more holistic study of knowledge evolution to elucidate complex biology.

Keywords: temporal graph, gene relationships, nomenclature evolution, interactive visualization, biological insights

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Authors: Rihab Ben Lamine, Raoudha Ben Jemaa, Ikram Amous Ben Amor

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This paper proposes a graph planning technique for semantic adaptable Web Services composition. First, we use an ontology based context model for extending Web Services descriptions with information about the most suitable context for its use. Then, we transform the composition problem into a semantic context aware graph planning problem to build the optimal service composition based on user's context. The construction of the planning graph is based on semantic context aware Web Service discovery that allows for each step to add most suitable Web Services in terms of semantic compatibility between the services parameters and their context similarity with the user's context. In the backward search step, semantic and contextual similarity scores are used to find best composed Web Services list. Finally, in the ranking step, a score is calculated for each best solution and a set of ranked solutions is returned to the user.

Keywords: semantic web service, web service composition, adaptation, context, graph planning

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Authors: I. Gahlouz, A. Chellil

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The paper concerns the estimation of system components faults by using an unknown inputs observer. To reach this goal, we used the Bond Graph approach to physical modelling. We showed that this graphical tool is allowing the representation of system components faults as unknown inputs within the state representation of the considered physical system. The study of the causal and structural features of the system (controllability, observability, finite structure, and infinite structure) based on the Bond Graph approach was hence fulfilled in order to design an unknown inputs observer which is used for the system component fault estimation.

Keywords: estimation, bond graph, controllability, observability

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Authors: Shin-Shin Kao, Hsiu-Chunj Pan

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Given a graph G. A cycle of G is a sequence of vertices of G such that the first and the last vertices are the same. A hamiltonian cycle of G is a cycle containing all vertices of G. The graph G is k-ordered (resp. k-ordered hamiltonian) if for any sequence of k distinct vertices of G, there exists a cycle (resp. hamiltonian cycle) in G containing these k vertices in the specified order. Obviously, any cycle in a graph is 1-ordered, 2-ordered and 3-ordered. Thus the study of any graph being k-ordered (resp. k-ordered hamiltonian) always starts with k = 4. Most studies about this topic work on graphs with no real applications. To our knowledge, the chordal ring families were the first one utilized as the underlying topology in interconnection networks and shown to be 4-ordered [1]. Furthermore, based on computer experimental results in [1], it was conjectured that some of them are 4-ordered hamiltonian. In this paper, we intend to give some possible directions in proving the conjecture.

Keywords: Hamiltonian cycle, 4-ordered, Chordal rings, 3-regular

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Authors: Sanju Vaidya, Aihua Li

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Vulnerability measures and topological indices are crucial in solving various problems such as the stability of the communication networks and development of mathematical models for chemical compounds. In 1947, Harry Wiener introduced a topological index related to molecular branching. Now there are more than 100 topological indices for graphs. For example, Hosoya polynomials (also called Wiener polynomials) were introduced to derive formulas for certain vulnerability measures and topological indices for various graphs. In this paper, we will find a relation between the Hosoya polynomials of any graph and its Mycielskian graph. Additionally, using this we will compute vulnerability measures, closeness and betweenness centrality, and extended Wiener indices. It is fascinating to see how Hosoya polynomials are useful in the two diverse fields, cybersecurity and chemistry.

Keywords: hosoya polynomial, mycielskian graph, graph vulnerability measure, topological index

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Authors: Wongsakorn Charoenpanitseri

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The total chromatic number χ"(G) of a graph G is the minimum number of colors needed to color the elements (vertices and edges) of G such that no incident or adjacent pair of elements receive the same color Let G be a graph with maximum degree Δ(G). Considering a total coloring of G and focusing on a vertex with maximum degree. A vertex with maximum degree needs a color and all Δ(G) edges incident to this vertex need more Δ(G) + 1 distinct colors. To color all vertices and all edges of G, it requires at least Δ(G) + 1 colors. That is, χ"(G) is at least Δ(G) + 1. However, no one can find a graph G with the total chromatic number which is greater than Δ(G) + 2. The Total Coloring Conjecture states that for every graph G, χ"(G) is at most Δ(G) + 2. In this paper, we prove that the Total Coloring Conjectur for a Δ-claw-free 3-degenerated graph. That is, we prove that the total chromatic number of every Δ-claw-free 3-degenerated graph is at most Δ(G) + 2.

Keywords: total colorings, the total chromatic number, 3-degenerated, CLAW-FREE

Procedia PDF Downloads 151