Kinds Of Graphs Research Articles

Conditional diagnosability of multiprocessor systems based on Cayley graphs generated by transpositions

Let Γn be the symmetric group on {1,2,…,n} and S be the generating set of Γn. The corresponding Cayley graph is denoted by Γn(S). If all elements of S are transpositions, a simple way to depict S is to via a graph, called the transposition generating graph of S, denoted by A(S) (or say simply A), where the vertex set of A is {1,2,…,n}, there is an edge in A between i and j if and only if the transposition (ij)∈S, and Γn(S) is called a Cayley graph obtained from a transposition generating graphA. Conditional diagnosability, proposed by Lai et al, is a novel measure of diagnosability that adds the additional condition that any faulty set cannot contain all of the neighbors of any vertex in a system. There are two well-known diagnostic models, PMC model and MM* model. The conditional diagnosability under the PMC (resp. MM*) model of a graph G, denoted by tcPMC(G) (resp. tcMM∗(G)), is the maximum value of t is the maximum value of t such that G is conditionally t-diagnosable under the PMC (resp. MM*) model. The conditional diagnosability of many well-known multiprocessor systems has been explored. In this paper, by exploring the structural properties of these Cayley graphs, suppose |E(A)| is the number of edges in the transposition generating graph A, we obtain the following results:(1) Under the MM* model, tcMM∗(Γn(S))=3|E(A)|−6,if A has a triangle;3|E(A)|−5,if A has no triangles and A is not a star.(2) Under the PMC model, tcPMC(Γn(S))=4|E(A)|−9,if A has a triangle;4|E(A)|−7,if A has no triangles and A is not a star.As corollaries, the conditional diagnosability of many kinds of graphs under the MM* model and the PMC model such as Cayley graphs generated by unicyclic graphs, wheel graphs, complete graphs, tree graphs etc. are obtained.

Darboux transformation and soliton solutions of a nonlocal Hirota equation

Starting from local coupled Hirota equations, we provide a reverse space-time nonlocal Hirota equation by the symmetry reduction method known as the Ablowitz–Kaup–Newell–Segur scattering problem. The Lax integrability of the nonlocal Hirota equation is also guaranteed by existence of the Lax pair. By Lax pair, an n-fold Darboux transformation is constructed for the nonlocal Hirota equation by which some types of exact solutions are found. The solutions with specific properties are distinct from those of the local Hirota equation. In order to further describe the properties and the dynamic features of the solutions explicitly, several kinds of graphs are depicted.

Total Proper Connection and Graph Operations

A graph is said to be total-colored if all the edges and vertices of the graph are colored. A path in a total-colored graph is a total proper path if (i) any two adjacent edges on the path differ in color, (ii) any two internal adjacent vertices on the path differ in color, and (iii) any internal vertex of the path differs in color from its incident edges on the path. A total-colored graph is called total-proper connected if any two vertices of the graph are connected by a total proper path of the graph. For a connected graph [Formula: see text], the total proper connection number of [Formula: see text], denoted by [Formula: see text], is defined as the smallest number of colors required to make [Formula: see text] total-proper connected. In this paper, we study the total proper connection number for the graph operations. We find that [Formula: see text] is the total proper connection number for the join, the lexicographic product and the strong product of nearly all graphs. Besides, we study three kinds of graphs with one factor to be traceable for the Cartesian product as well as the permutation graphs of the star and traceable graphs. The values of the total proper connection number for these graphs are all [Formula: see text].

Result Involution Graphs of Finite Groups

In this paper, a new kind of graph on a finite group , namely the result involution graph is defined and studied. We use to denote this graph, is a simple undirected graph with vertex set. Two distinct vertices are adjacent if and only if their product is nontrivial involution element in . The result involution graph for several finite groups are obtained. We study some properties of the result involution graph by resizing graph by using the conjugacy classes of . Finally, we show that the result involution graphs for the symmetric groups and the alternating groups are connected with diameter at most 3 and radius at most 2 for. Furthermore, they have girth 3.

The controllability of graphs with diameter [formula omitted

Controllable graphs are connected graphs in which all eigenvalues are mutually distinct and main. In this work, a new method of characterizing the controllability of graphs with diameter n−2 is presented. A necessary and sufficient condition determining non-main eigenvalue of graphs with diameter n−2 is obtained, and the controllability of two kinds of graphs with diameter n−2 is characterized. Besides, the visualization representation of statistical results of controllable graphs is presented, and they show that the proportion of controllable graphs among the graphs with diameter n−2 is stablely at 15%, which partly verifies a conjecture proposed by Stanić.

A Quasi-Hole Detection Algorithm for Recognizing k-Distance-Hereditary Graphs, with k < 2

Cicerone and Di Stefano defined and studied the class of k-distance-hereditary graphs, i.e., graphs where the distance in each connected induced subgraph is at most k times the distance in the whole graph. The defined graphs represent a generalization of the well known distance-hereditary graphs, which actually correspond to 1-distance-hereditary graphs. In this paper we make a step forward in the study of these new graphs by providing characterizations for the class of all the k-distance-hereditary graphs such that k<2. The new characterizations are given in terms of both forbidden subgraphs and cycle-chord properties. Such results also lead to devise a polynomial-time recognition algorithm for this kind of graph that, according to the provided characterizations, simply detects the presence of quasi-holes in any given graph.

Calculating the Number of vertices Labeled Order Six Disconnected Graphs which Contain Maximum Seven Loops and Even Number of Non-loop Edges Without Parallel Edges

A labeled graph is a graph where each vertex or edge is given a value or label. A line/edge with the same starting and ending vertex is called a loop. If given n vertices and m edges, then there are lot disconnected vertices labeled graphs that can be formed. Among those graphs, there are graphs with maximum seven loops and whose non-loop edges are even. In this research, we will discuss the formula for finding that kinds of graphs.

Index-Based Intimate-Core Community Search in Large Weighted Graphs

Community search that finds query-dependent communities has been studied on various kinds of graphs. As one instance of community search, intimate-core group (community) search over a weighted graph is to find a connected <inline-formula><tex-math notation="LaTeX">$k$</tex-math></inline-formula> -core containing all query nodes with the smallest group weight. However, existing state-of-the-art methods start from the maximal <inline-formula><tex-math notation="LaTeX">$k$</tex-math></inline-formula> -core to refine an answer, which is practically inefficient for large networks. In this paper, we develop an efficient framework, called <u>l</u> ocal <u>e</u> xploration <u>k</u> -core <u>s</u> earch (LEKS), to find intimate-core groups in graphs. We propose a small-weighted spanning tree to connect query nodes, and then expand the tree level by level to a connected <inline-formula><tex-math notation="LaTeX">$k$</tex-math></inline-formula> -core, which is finally refined as an intimate-core group. In addition, to support the intimate group search over large weighted graphs, we develop a weighted-core index (WC-index) and two new WC-index-based algorithms for expansion and refinement phases in LEKS. Specifically, we propose a WC-index-based expansion to efficiently find a candidate graph of intimate-core group, leveraging on a two-level expansion of <inline-formula><tex-math notation="LaTeX">$k$</tex-math></inline-formula> -breadth and 1-depth. We propose two graph removal strategies: coarse-grained refinement is designed for large graphs to delete a batch of nodes in a few iterations; fine-grained refinement is designed for small graphs to remove nodes carefully and achieve high-quality answers. Extensive experiments on real-life networks with ground-truth communities validate the effectiveness and efficiency of our proposed methods.

GS4: Graph stream summarization based on both the structure and semantics

Nowadays internet-based applications collect and distribute large datasets, which are mostly modeled by pertinent massive graphs. One solution to process such massive graphs is summarization. There are two kinds of graphs, stationary and stream. There are several algorithms to summarize stationary graphs; however, no comprehensive method has been devised to summarize stream graphs. This is because of the challenges of the graph stream, which are the high data volume and the continuous changes of data over time. To tackle such challenges, we propose a novel method based on the sliding window model that performs summarization using both the structure and vertex attributes of the input graph stream. We devise a new structure for a summary graph by considering the structural and semantical attributes that can better elucidate every heterogeneous summary graph. Moreover, our framework comprises innovative components for comparing hybrid summary graphs. To the best of our knowledge, this is the first method that summarizes a graph stream using both the structure and vertex attributes with varying contributions. Our approach also takes user directions and ontology into account. Aiming to study the efficiency and effectiveness of our proposed method, we conduct extensive experiments on two real-life datasets: American political web-logs and Amazon co-purchasing products. The experimental results confirm that compared to the existing approaches the proposed method generates graph summaries with better quality. The expected time of our proposed method in this paper ( $$O(n^3)$$ ) has significantly enhanced the efficiency compared to the current best complexity which is $$O(n^5)$$ .

Ideal Graphs Supported By Given Ideals of Commutative Rings

In this paper we introduce and study a new kind of graph that constructed by non-trivial ideals of a commutative ring with identity. Let R be a commutative ring with identity and P be a non-trivial ideal of R. The ideal graph supported by the ideal P, denoted by (P), is the undirected graph whose vertices are those non-trivial ideals I of R such that there exists a non-trivial ideal JI of R with IJ⊂P, and every two vertices I and J are adjacent if IJ and IJ⊂P. We investigate the connectivity, completeness and planarity of the graph (P). Also we explore the diameter, girth, domination, clique number and chromatic number of (P).

On graphs with adjacency and signless Laplacian matrices eigenvectors entries in {−1,+1}

In 1986, Herbert Wilf asked what kind of graphs have an eigenvector with entries formed only by ±1. In this paper, we answer this question for the adjacency, Laplacian, and signless Laplacian matrices of a graph. To this end, we generalize the concept of an exact graph to the adjacency and signless Laplacian matrices. Infinite families of exact graphs for all those matrices are presented.

Tensor Graph Convolutional Networks for Text Classification

Compared to sequential learning models, graph-based neural networks exhibit some excellent properties, such as ability capturing global information. In this paper, we investigate graph-based neural networks for text classification problem. A new framework TensorGCN (tensor graph convolutional networks), is presented for this task. A text graph tensor is firstly constructed to describe semantic, syntactic, and sequential contextual information. Then, two kinds of propagation learning perform on the text graph tensor. The first is intra-graph propagation used for aggregating information from neighborhood nodes in a single graph. The second is inter-graph propagation used for harmonizing heterogeneous information between graphs. Extensive experiments are conducted on benchmark datasets, and the results illustrate the effectiveness of our proposed framework. Our proposed TensorGCN presents an effective way to harmonize and integrate heterogeneous information from different kinds of graphs.

Agglomerative community detection on signed networks based on community adjacency list

With the development of data processing technology, complex network theory has been widely applied in many areas. Meanwhile, as one of the essential parts of network science, community detection is becoming more and more important for analyzing and visualizing the real world. Specially, signed network is a kind of graph which can more truly and efficiently reflect the reality, however, the study of community detection on signed network is still rare. In this paper, we propose a new agglomerative algorithm based on the modularity optimization for community detection on signed networks. The proposed model utilizes a new data structure called community adjacency list in signed (CALS) networks to improve the efficiency. Successive modularity computations make the connections between node changes so that the process time leads to substantial savings. Experiments on both real and artificial networks verify the accuracy and efficiency of this method, which is suitable for the application on large-scale networks.

Squareness for the Monopole-Dimer Model

The monopole-dimer model introduced recently is an exactly-solvable signed generalisation of the dimer model. We show that the partition function of the monopole-dimer model on a graph invariant under a fixed-point free involution is a perfect square. We give a combinatorial interpretation of the square-root of the partition function for such graphs in terms of a monopole-dimer model on a new kind of graph with two types of edges which we call a dicot. The partition function of the latter can be written as a determinant, this time of a complex adjacency matrix. This formulation generalises T. T. Wu's assignment of imaginary orientation for the grid graph to planar dicots. As an application, we compute the partition function for a family of non-planar dicots with positive weights.

Overlap graphs and de Bruijn graphs: data structures for de novogenome assembly in the big data era

BackgroundDe novo genome assembly relies on two kinds of graphs: de Bruijn graphs and overlap graphs. Overlap graphs are the basis for the Celera assembler, while de Bruijn graphs have become the dominant technical device in the last decade. Those two kinds of graphs are collectively called assembly graphs.ResultsIn this review, we discuss the most recent advances in the problem of constructing, representing and navigating assembly graphs, focusing on very large datasets. We will also explore some computational techniques, such as the Bloom filter, to compactly store graphs while keeping all functionalities intact.ConclusionsWe complete our analysis with a discussion on the algorithmic issues of assembling from long reads ( e.g., PacBio and Oxford Nanopore). Finally, we present some of the most relevant open problems in this field.

Sum Ideal Graphs Associated to a Given Ideal of a Commutative Ring

The aim of this paper is to introduce and study a new kind of graphs associated to an ideal of a commutative ring. Let ℛ be a commutative ring with identity, and I(ℛ) be the set of all non-trivial ideals of ℛ with S I(ℛ). The sum ideal graph associated to S, denoted by Ψ(ℛ, S), is the undirected graph with vertex set {A I(ℛ): S⊂A+B, for some B I(ℛ)} where two ideal vertices A and B are adjacent if and only if A B and S⊂A+B. In this paper we establish some of characterizations and results of this kind of graph with providing some examples.

A new method for graph stream summarization based on both the structure and concepts

Abstract Graph datasets are common in many application domains and for which their graphs are usually massive. One solution to process such massive graphs is summarization. There are two kinds of graphs, stationary and stream. For stationary graphs, a number of summarization algorithms are available while for graph stream there is no a comprehensive summarization method that summarizes a graph stream based on the structure, vertex attributes or both with varying contributions. This is because of challenges of graph stream, which are volume of data and changing of data over time. In this paper, we propose a method based on sliding-window model for which summarizes a graph stream based on a combination of the structure and vertex attributes. We proposed a new structure for summary graphs and also proposed new methods for comparing two summary graphs. To the best of our knowledge, this is the first method that summarizes a graph stream based on both the structure and vertex attributes with varying contributions. Through extensive experiments on real dataset of Amazon co-purchasing products, we have demonstrated the performance of the proposed method.

Clustering Structure-Induced Robust Multi-View Graph Recovery

Graph based classification methods have been widely applied in the fields of computer vision and machine learning. The quality of the graph highly affects the performance of these methods. The same object is commonly represented by different features, i.e., multi-view features, which leads to multiple graphs corresponding to different features in multi-view learning. However, what kind of graph is important for the task is unknown in advance. Moreover, existing multi-view learning methods become weak in dealing with noisy graphs when the data is corrupted by the noise. In this paper, we address this problem by observing that the noise of each graph has specific structure. Then, based on this observation we propose a robust multi-view graph recovery (RMGR) method in which the specific structure is used to clean the multiple input noisy graphs and these cleaned graphs are simultaneously aggregated into a consensus graph by adaptively assigning great weighted coefficients for important graphs. To make the consensus graph suit classification, the clustering structure is introduced to restrain the rank of Laplacian matrix of the consensus graph such that the number of its connected components is equal to that of clustering. In doing so, the graph is adaptively adjusted during optimization to more accurately partition data. The optimization problem is solved by proposed the iterative update algorithm. Extensive experiments on synthetic and several benchmark data sets show the effectiveness of the proposed method.

Strongly Menger connectedness of data center network and (n,k)-star graph

A graph G=(V,E) is F-strongly Menger-vertex-connected (resp. Menger-edge-connected), if for a subgraph G−F of G with a conditional faulty vertex (resp. edge) set F⊆V (resp. F⊆E), each pair of vertices u and v are connected by min{degG−F(u),degG−F(v)} internally disjoint (resp. edge-disjoint) fault-free paths in G−F. Where degG−F(u) and degG−F(v) are the degrees of u and v in G−F, respectively. He et al. (2018) [7] introduced some sufficient conditions of F-strongly Menger (edge) connected for k-regular triangle-free graphs. In this paper, we consider the strongly Menger connectedness of two kinds of graphs with triangles: data center network Dk,n and (n,k)-star graph Sn,k. We prove Dk,n is (k−1)-strongly Menger-vertex-connected, (n+k−3)-strongly Menger-edge-connected of order 1, (2n+2k−8)-strongly Menger-edge-connected of order 2, and (3n+3k−15)-strongly Menger-edge-connected of order 3. Furthermore, we obtain Sn,k is (k−2)-strongly Menger-vertex-connected, (n−3)-strongly Menger-edge-connected of order 1, (2n−8)-strongly Menger-edge-connected of order 2, and (3n−15)-strongly Menger-edge-connected of order 3.

Algorithmic implementation of signed product and total signed product cordial labeling on complex graph

Apart from other labeling signed product and total signed product labeling already have been applied on different kind of graphs. Signed product cordial labeling for the graph G = (V , E ), where each vertex is labelled by 1 or -1 and edge label is