Edge Labels Research Articles

An Efficient Algorithm for Embedding Two-Dimensional Tori into Balanced Hypercubes

Balanced hypercube is a variant of the hypercube structure; it offers desirable properties such as connectivity, diameter, and fault-tolerance. Cycles and tori are popular interconnection topologies and have been widely used in distributed-memory parallel computers. Scholars have developed diverse and numerous parallel algorithms of cycles and tori. This paper proposes an efficient algorithm, called reflected edge label sequence, to generate cycle structures in balanced hypercubes. With this algorithm, we can embed cycles of length [Formula: see text], [Formula: see text], into an [Formula: see text]-dimensional balanced hypercube. The corresponding algorithm for embedding 2-dimensional [Formula: see text] tori can be constructed on the basis of this approach, and the time complexity is linear with respect to the size of a balanced hypercube.

Application of the Combinatorial Nullstellensatz to Integer-magic Graph Labelings

Let $A$ be a nontrivial abelian group and $A^* = A \setminus \{0\}$. A graph is $A$-magic if there exists an edge labeling $f$ using elements of $A^*$ which induces a constant vertex labeling of the graph. Such a labeling $f$ is called an $A$-magic labeling and the constant value of the induced vertex labeling is called an $A$-magic value. In this paper, we use the Combinatorial Nullstellensatz to show the existence of $\mathbb{Z}_p$-magic labelings (prime $p \geq 3$ ) for various graphs, without having to construct the $\mathbb{Z}_p$-magic labelings. Through many examples, we illustrate the usefulness (and limitations) in applying the Combinatorial Nullstellensatz to the integer-magic labeling problem. Finally, we focus on $\mathbb{Z}_3$-magic labelings and give some results for various classes of graphs.

United Monoids

Graphs and various graph-like combinatorial structures, such as preorders and hypergraphs, are ubiquitous in programming. This paper focuses on representing graphs in a purely functional programming language like Haskell. There are several existing approaches; one of the most recently developed ones is the "algebraic graphs" approach (2017). It uses an algebraic data type to represent graphs and has attracted users, including from industry, due to its emphasis on equational reasoning and making a common class of bugs impossible by eliminating internal invariants. The previous formulation of algebraic graphs did not support edge labels, which was a serious practical limitation. In this paper, we redesign the main algebraic data type and remove this limitation. We follow a fairly standard approach of parameterising a data structure with a semiring of edge labels. The new formulation is both more general and simpler: the two operations for composing graphs used in the previous work can now be obtained from a single operation by fixing the semiring parameter to zero and one, respectively. By instantiating the new data type with different semirings, and working out laws for interpreting the resulting expression trees, we discover an unusual algebraic structure, which we call "united monoids", that is, a pair of monoids whose unit elements coincide. We believe that it is worth studying united monoids in their full generality, going beyond the graphs which prompted their discovery. To that end, we characterise united monoids with a minimal set of axioms, prove a few basic theorems, and discuss several notable examples. We validate the presented approach by implementing it in the open-source *algebraic-graphs* library. Our theoretical contributions are supported by proofs that are included in the paper and have also been machine-checked in Agda. By extending algebraic graphs with support for edge labels, we make them suitable for a much larger class of possible applications. By studying united monoids, we provide a theoretical foundation for further research in this area.

Mask-Guided Deformation Adaptive Network for Human Parsing

Due to the challenges of densely compacted body parts, nonrigid clothing items, and severe overlap in crowd scenes, human parsing needs to focus more on multilevel feature representations compared to general scene parsing tasks. Based on this observation, we propose to introduce the auxiliary task of human mask and edge detection to facilitate human parsing. Different from human parsing, which exploits the discriminative features of each category, human mask and edge detection emphasizes the boundaries of semantic parsing regions and the difference between foreground humans and background clutter, which benefits the parsing predictions of crowd scenes and small human parts. Specifically, we extract human mask and edge labels from the human parsing annotations and train a shared encoder with three independent decoders for the three mutually beneficial tasks. Furthermore, the decoder feature maps of the human mask prediction branch are further exploited as attention maps, indicating human regions to facilitate the decoding process of human parsing and human edge detection. In addition to these auxiliary tasks, we further alleviate the problem of deformed clothing items under various human poses by tracking the deformation patterns with the deformable convolution. Extensive experiments show that the proposed method can achieve superior performance against state-of-the-art methods on both single and multiple human parsing datasets. Codes and trained models are available https://github.com/ViktorLiang/MGDAN .

Edge Irregular Reflexive Labeling for Some Classes of Plane Graphs

For a graph G, we define a total k-labeling ϕ as a combination of an edge labeling ϕe(x) → {1, 2, . . . , ke} and a vertex labeling ϕv(x) → {0, 2, . . . , 2kv}, such that ϕ(x) = ϕv(x) if x ∈ V (G) and ϕ(x) = ϕe(x) if x ∈ E(G), where k = max {ke, 2kv}. The total k-labeling ϕ is called an edge irregular reflexive k-labeling of G, if for every two edges xy, x0y0of G, one has wt(xy) 6=wt(x0y0), where wt(xy) = ϕv(x) + ϕe(xy) + ϕv(y). The smallest value of k for which such labeling exists is called a reflexive edge strength of G. In this paper, we study the edge irregular reflexive labeling on plane graphs and determine its reflexive edge strength.

Module Basis for Generalized Spline Modules

Let G = (V,E) be a graph of order n. Let R be a commutative ring and I denote the set of all ideals of R. Let ? : E ? I be an edge labeling. A generalized spline of (G, ?) is a vertex labeling F : V ? R such that for each edge uv, F(u) ? F(v) ? ?(uv). The set R(G,) of all generalized splines of (G, ?) is an R-module. In this paper we determine conditions for a subset of R(G,?) to form a basis of R(G,?) for some classes of graphs.

Flexibility and Movability in Cayley Graphs

Let \({\varvec{\Gamma }} = (V,E)\) be a (non-trivial) finite graph with \(\lambda : E \rightarrow {\mathbb {R}}_{+}\) an edge labeling of \({\varvec{\Gamma }}\). Let \(\rho : V\rightarrow {\mathbb {R}}^{2}\) be a map which preserves the edge labeling, i.e., $$\begin{aligned} \Vert \rho (u) - \rho (v)\Vert _{2} = \lambda ((u,v)), \,\forall (u,v)\in E, \end{aligned}$$where \(\Vert x-y\Vert _{2}\) denotes the Euclidean distance between two points \(x,y \in {\mathbb {R}}^{2}\). The labeled graph is said to be flexible if there exists an infinite number of such maps (up to equivalence by rigid transformations) and it is said to be movable if there exists an infinite number of injective maps (again up to equivalence by rigid transformations). We study movability of Cayley graphs and construct regular movable graphs of all degrees. Further, we give explicit constructions of dense, movable graphs.

On some graph-cordial Abelian groups

Hovey introduced A-cordial labelings as a generalization of cordial and harmonious labelings [11]. If A is an Abelian group, then a labeling f:V(G)→A of the vertices of some graph G induces an edge labeling on G; the edge uv receives the label f(u)+f(v). A graph G is A-cordial if there is a vertex-labeling such that (1) the vertex label classes differ in size by at most one and (2) the induced edge label classes differ in size by at most one.In the literature mostly a cordial labeling in cyclic groups is studied. Patrias and Pechenik studied the larger class of finite Abelian groups A. They posed a conjecture that for every group A there is an A-cordial labeling for almost every path. In this paper we solve this conjecture. Moreover we show that all cycle graphs are A-cordial for any Abelian group A of odd order.

Graph polynomials and group coloring of graphs

Let Γ be an Abelian group and let G be a simple graph. We say that G is Γ-colorable if for some fixed orientation of G and every edge labeling ℓ:E(G)→Γ, there exists a vertex coloring c by the elements of Γ such that c(y)−c(x)≠ℓ(e), for every edge e=xy (oriented from x to y).Langhede and Thomassen proved recently that every planar graph on n vertices has at least 2n/9 different Z5-colorings. By using a different approach based on graph polynomials, we extend this result to K5-minor-free graphs in the more general setting of field coloring. More specifically, we prove that every such graph on n vertices is F-5-choosable, whenever F is an arbitrary field with at least 5 elements. Moreover, the number of colorings (for every list assignment) is at least 5n/4.

Region-Level SAR Image Segmentation Based on Edge Feature and Label Assistance

This paper proposes a novel segmentation algorithm for synthetic aperture radar (SAR) images. The algorithm performs region-level segmentation based on edge feature and label assistance (REFLA). It demonstrates improved performance in terms of segmentation accuracy while better preserving image edges. Firstly, an edge detection scheme is implemented, which fuses information from two advanced edge detection methods, thereby obtaining a more precise edge strength map (ESM). Secondly, a Canny algorithm is performed to divide the SAR image into edge regions and homogeneous regions, and different smoothing templates are selected according to pixel positions. Therefore, an anisotropic smoothing on the SAR image can be achieved, aiming at suppressing the noise within targets while also accurately maintaining the target boundaries. Thirdly, K-means clustering is applied on the smoothed result, to generate an initial set of labels. Using ESM and the initial labels as inputs, a watershed transformation and a majority voting strategy are employed to realize an initial segmentation at the region level. Finally, a label-aided region merging (LaRM) strategy is used to correctly segment the wrongly labeled regions, to give the final segmentation result. The LaRM, with merging rules based on label rather than gray characteristics, can avoid the need for calculating a large number of complex formulae, thus accelerating the region merging. Results are presented of experiments, on both simulated and real SAR images, in which the proposed REFLA method is compared against six state-of-the-art algorithms from the literature. REFLA achieves higher accuracy, while better retaining the image edges.

On Edge Irregular Reflexive Labeling for Generalized Prism

Among the various ideas that appear while studying graph theory, which has gained much attraction especially in graph labeling, labeling of graphs gives mathematical models which value for a vast range of applications in high technology (data security, cryptography, various problems of coding theory, astronomy, data security, telecommunication networks, etc.). A graph label is a designation of graph elements, i.e., the edges and/or vertex of a group of numbers (natural numbers), and is called assignment or labeling. The vertex or edge labeling is related to their domain asset of vertices or edges. Likewise, for total labeling, we take the domain as vertices and edges both at the same time. The reflexive edge irregularity strength (res) is total labeling in which weights of edges are not the same for all edges and the weight of an edge is taken as the sum of the edge labels and the vertices associated with that edge. In the res, the vertices are labeled with nonnegative even integers while the edges are labeled with positive integers. We have to make the labels minimum, whether they are associated with vertices or edges. If such labeling exists, then it is called the res of H and is represented as s res(H). In this paper, we have computed the res for the Cartesian product of path and cycle graph which is also known as generalizing prism.

Some Types of Irregular Labeling of Diamond Networks on Ten Vertices

There are three interesting parameters in irregular networks based on total labelling, i.e. the total vertex irregularity strength, the total edge irregularity strength, and the total irregularity strength of a graph. Besides that, there is a parameter based on edge labelling, i.e., the irregular labelling. In this paper, we determined the four parameters for diamond graph on eight vertices.

Extracting malicious behaviours

In recent years, the damage cost caused by malwares is huge. Thus, malware detection is a big challenge. The task of specifying malware takes a huge amount of time and engineering effort since it currently requires the manual study of the malicious code. Thus, in order to avoid the tedious manual analysis of malicious codes, this task has to be automatised. To this aim, we propose in this work to represent malicious behaviours using extended API call graphs, where nodes correspond to API function calls, edges specify the execution order between the API functions, and edge labels indicate the dependence relation between API functions parameters. We define new static analysis techniques that allow to extract such graphs from programs, and show how to automatically extract, from a set of malicious and benign programs, an extended API call graph that represents the malicious behaviours. Finally, we show how this graph can be used for malware detection. We implemented our techniques and obtained encouraging results: 95.66% of detection rate with 0% of false alarms.

A New Notion of Classical Mean Graphs Based on Duplicating Operations

Classical mean labeling of a graph G with p vertices and q edges is an injective function from the vertex set of G to the set {1,2,3, …, q + 1} such that the edge labels obtained from the flooring function of the average of mean of arithmetic, geometric, harmonic, and root square of the vertex labels of each edge’s end vertices is distinct, and the set of edge labels is {1,2,3, …, q}. One of the graph operations is to duplicate the graph. The classical meanness of graphs formed by duplicating an edge and a vertex of numerous standard graphs is discussed in this study.

Edge-Inferring Graph Neural Network With Dynamic Task-Guided Self-Diagnosis for Few-Shot Hyperspectral Image Classification

The current hyperspectral image classification (HSIC) model based on the convolutional neural network for feature extraction and softmax classifier has been prone to the barrier of label prediction with limited samples. Substituting for the enormously complicated work of terrain labeling, few-shot learning provides a popular option for HSIC with very few annotated samples. In this paper, we proposed a novel edge-inferring framework with the meta-learning paradigm for hyperspectral few-shot classification (HSFSC). In which, a graph neural network for similarity measurement is firstly presented to iteratively infer edge labels with the exploitation of instance-level similarity and the distribution-level similarity. Besides, in the meta-training stage, the pixel prediction model and patch prediction model based on edge inferring architecture are concretized jointly to improve the classification accuracy of the test samples. Expressly, at the meta-testing phase, the dynamic task-guided self-diagnosis strategy is developed for the first time to diagnose the samples separability of the current classification task, which is responsible for dynamically assigning the most reliable results based on the generated reliability grade of the sample. The extensive experimental results and analysis of three hyperspectral image datasets demonstrate the superiority of the proposed HSFSC architecture compared with other advanced methods.

Tool wear condition monitoring method based on graph neural network with a single sensor

Tool wear condition monitoring (TCM) is an important part of machining automation. In recent years, deep learning (DL)-based TCM methods have been widely researched. However, almost all DL-based methods need sufficient learning samples to obtain good accuracy, which is hard for TCM in terms of cost and time. In order to enhance the recognition accuracy of DL-based TCM under small samples, this paper proposed a new improved multi-scale edge-labelling graph neural network (MEGNN). Firstly, the signal of cutting force sensor is expanded to multi-dimensional data through phase space reconstruction. Secondly, these multi-dimensional data are encoded into a recurrence plot (RP). Then, the RP is input to multi-scale EGNN (MEGNN) to extract features. Finally, the tool condition is estimated through the updated edge labels using a weighted voting method. Our milling TCM experiments demonstrate the proposed MEGNN-based TCM method outperforms three DL-based methods under small samples.

The integer-antimagic spectra of a disjoint union of Hamiltonian graphs

Let $A$ be a nontrivial abelian group. A simple graph $G = (V, E)$ is $A$-antimagic, if there exists an edge labeling $f: E(G) \to A \backslash \{0\}$ such that the induced vertex labeling $f^+(v)=\sum_{uv\in E(G)} f(uv)$ is a one-to-one map. The {integer-antimagic spectrum} of a graph $G$ is the set IAM$(G) = \{k: G {is} \mathbb{Z}_k{-antimagic and } k \geq 2\}$. In this paper, we determine the integer-antimagic spectra for a disjoint union of Hamiltonian graphs.

Odd graceful labeling of super subdivision of few graphs

Odd graceful labeling was originally introduced by Gnanajothi [3] in 1991. The graph Gdemonstrates Odd graceful labeling as injecting β from V(G) → {0,1,2,… (2q - 1)} in such a way that whenever the label |β(x) - β(y)| is allocated to every edgexy, the derived edge labels are {1,3,5,…, (2q - 1). This paper demonstrates that the Super subdivision of the star and super subdivision of path permits odd graceful labeling.

FSL-EGNN: Edge-Labeling Graph Neural Network for Hyperspectral Image Few-Shot Classification

The existing hyperspectral image (HSI) classification encounters the obstacle of improving the classification accuracy with limited labeled samples. In this context, as a typical implementation of meta-learning, few-shot learning (FSL) makes the model learn by episodic training on source HSI, which has achieved significant improvements in small sample classification of target HSI. However, the existing FSL methods lack explicit consideration and exploration of the association between pixels, especially the intraclass association and interclass association between pixels in the support set and query set. To mitigate these issues, an FSL method based on edge-labeling graph neural network (FSL-EGNN) is proposed for small sample classification of HSI, which is the first attempt to explicitly quantify the associations between pixels by exploiting EGNN in HSI few-shot classification (FSC). Specifically, based on graph construction of HSI, episodic training is performed on the existing source HSI. During training, EGNN is used to predict the edge labels on the graph, thereby explicitly modeling the intraclass similarity and interclass dissimilarity between pixels of HSI. After the trained model is fine-tuned, it can realize FSC on the unseen target HSI. Experiments conducted on three benchmark HSI datasets demonstrate that the proposed FSL-EGNN outperforms the existing state-of-the-art methods with limited labeled samples.

Strongly Multiplicative Labeling of Diamond Graph, Generalized Petersen Graph, and Some Other Graphs

A finite, simple graph of order k is said to be a strongly multiplicative graph when all vertices of the graph are labeled by positive integers 1,2,3, …, k such that the induced edge labels of the graph, obtained by the product of labels of end vertices of edges, are distinct. In this paper, we show that the diamond graph Brn for 𝑛 ≥ 3, umbrella graph Um,n, and generalized Petersen graph GP(n, k), for n ≥ 3 and 1 ≤ k < (n/2), admit strongly multiplicative labeling. Moreover, strongly multiplicative labeling of a double comb graph and sunflower planar graph has also been investigated and elaborated as well with different examples.