Labeling Of Graph Research Articles

Solving the minimum labeling global cut problem by mathematical programming

AbstractAn edge‐labeled graph (ELG) is a graph such that is the set of vertices, is the set of edges, is the set of labels (colors), and each edge has a label associated. Given an ELG , the goal of the minimum labeling global cut problem (MLGCP) is to find a subset such that the removal of all edges with labels in disconnects and is minimum. This work proposes three new mathematical formulations for the MLGCP, namely PART, VC, and TE as well as branch‐and‐cut algorithms to solve them. Additionally, a theoretical study was carried out on the MLGCP input graph, leading to the concept of chromatic closure, used in preprocessing algorithms for this model PART. Finally, a comprehensive polyhedral investigation of the model is performed. The computational experiments showed that the model, adopting the chromatic closure concept and its branch‐and‐cut algorithm, can solve small to average‐sized instances in reasonable times.

Sturgeon-MKIV: Constraint-Based Level and Playthrough Generation with Graph Label Rewrite Rules

Procedurally generated game levels should be completable. The representation used for levels and game mechanics impacts the types of games for which different techniques can be applied. Previous work used a constraint solving approach to simultaneously generate levels with example playthroughs, showing they can be completed using the game's mechanics. However, that work used 2D grid-based rewrite rules. In this work, we extend previous approaches by representing levels as more general graphs, and game mechanics as rewrites on node and edge labels of subgraphs. Using this approach, graph-based levels with playthroughs are generated. We describe the approach and demonstrate its application in some games with graph-based levels.

Binary vertex labelings of graphs and digraphs

A (0; 1)-labeling of a set is said to be friendly if the number of elements of the set labeled 0 and the number labeled 1 differ by at most 1. Let g be a labeling of the edge set of a graph that is induced by a labeling f of the vertex set. If both g and f are friendly then f is said to be a cordial labeling of the graph. This concept extended to directed graphs is called (2; 3)-cordiality of digraphs. We investigate the labelings that are both cordial for a graph and (2; 3)-cordial for an orientation of it. We also consider the same problem for other known binary vertex labelings of graphs.

Pair mean cordial labeling of udukkai, octopus, drum and fire cracker graphs

In this paper, we investigate the pair mean cordial labeling behavior of udukkai graph, octopus graph, double wheel graph, barbell graph, drum graph, vanessa graph and fire cracker graph.

ZONAL LABELING OF EDGE COMB PRODUCT OF GRAPHS

Given a plane graph $G=(V,E)$. A zonal labeling of graph $G$ is defined as an assignment of the two nonzero elements of the ring $\mathbb{Z}_3$, which are $1$ and $2$, to the vertices of $G$ such that the sum of the labels of the vertices on the border of each region of the graph is $0\in\mathbb{Z}_3$. A graph $G$ that possess such a labeling is termed as zonal graph. This paper will characterize edge comb product graphs that are zonal. The results show that $P_m\trianglerighteq_eC_n$, $C_n\trianglerighteq_e C_r$, $S_p\trianglerighteq_e C_n$, and $S_p\trianglerighteq_e F_t$ are zonal in some cases, but not in others.

Elegant Fuzzy Labeling of Graphs

Fuzzy graphs and fuzzy labelings are crucial in modeling real-time systems by accommodating varying degrees of information precision. In this paper, we introduce a novel fuzzy labeling called Elegant Fuzzy Labeling. We prove that a simple graph admits Elegant Fuzzy Labeling if and only if it admits elegant labeling. Furthermore, we prove that while a simple graph that admits Elegant Fuzzy Labeling also admits fuzzy labeling, but not conversely. We identify specific classes of simple graphs that admit Elegant Fuzzy Labeling and explore practical application of this new labeling approach.

Zero-sum partitions of Abelian groups and their applications to magic- and antimagic-type labelings

The following problem has been known since the 80s. Let $\Gamma$ be an Abelian group of order $m$ (denoted $|\Gamma|=m$), and let $t$ and $\{m_i\}_{i=1}^{t}$, be positive integers such that $\sum_{i=1}^t m_i=m-1$. Determine when $\Gamma^*=\Gamma\setminus\{0\}$, the set of non-zero elements of $\Gamma$, can be partitioned into disjoint subsets $\{S_i\}_{i=1}^{t}$ such that $|S_i|=m_i$ and $\sum_{s\in S_i}s=0$ for every $1 \leq i \leq t$. Such a subset partition is called a \textit{zero-sum partition}. $|I(\Gamma)|\neq 1$, where $I(\Gamma)$ is the set of involutions in $\Gamma$, is a necessary condition for the existence of zero-sum partitions. In this paper, we show that the additional condition of $m_i\geq 4$ for every $1 \leq i \leq t$, is sufficient. Moreover, we present some applications of zero-sum partitions to magic- and antimagic-type labelings of graphs.

THE REFLEXIVE EDGE STRENGTH OF THE PENTAGONAL SNAKE GRAPH AND CORONA OF THE OPEN TRIANGULAR LADDER AND NULL GRAPH

Assume that be an undirected simple graph with vertex set and edge set . The edge irregular reflexive -labeling of graph is a labeling selects positive integers from 1 to as edge labels and non negative even numbers from 0 to as vertex labels, and the weights assigned to each edge are distinct, where . On graph with labeling, the weight of edge is represented by which is defined as the sum of edge label and all vertex labels incident to that edge. Reflexive edge strength of graph is the minimum of the highest label, denoted by . In this research, reflexive edge strength for pentagonal snake graph and corona of open triangular ladder and null graph will be determined. The method of this research is literature study, the lower bound of determined by Ryan’s lemma and the upper bound by labeling. The reflexive edge strength of pentagonal snake graph with is for and for The reflexive edge strength of corona of open triangular ladder and null graph with n ≥ 3 and m ≥ 1 is and .

PAIR MEAN CORDIAL LABELING OF HURDLE, KEY, LOTUS, AND NECKLACE GRAPHS

Let be a graph with vertices and edges. Define and . Consider a mapping by assigning different labels in to the different elements of when is even and different labels in to elements of V and repeating a label for the remaining one vertex when is odd. The labeling as defined above is said to be a pair mean cordial labeling if for each edge of G, there exists a labeling if is even and if is odd such that | |≤1 where and respectively denote the number of edges labeled with 1 and the number of edges not labeled with 1. A graph G for which there is a pair mean cordial labeling is called pair mean cordial graph(PMC-graph). In this paper, we investigate the pair mean cordial labeling of some graphs like hurdle graph, lotus graph, necklace graph, F-tree, Y-tree, subdivided shell graph, uniform bow graph and key graph.

Fuzzy Range Labeling of Fuzzy Star Graph and Helm Graph

Objectives: To find a new labeling technique in Fuzzy Graph Theory called Fuzzy Range Labeling. This concept helps to handle the uncertainty in graph labeling areas. Methods: Fuzzy graph labeling condition - and Range labeling condition - (that is, the edge membership value is calculated as the difference between the maximum and minimum values of the vertex incident to that edge) are used. Findings: In this study, we have found that the fuzzy star graph and helm graph are fuzzy range graphs. Novelty: The fuzzy range labeling technique was not yet studied. Thus, this new notion will motivate many researchers to study further. Keywords: Fuzzy graph labeling, Range labeling, Fuzzy range labeling, Fuzzy range graph, Fuzzy star graph, Helm graph

Arithmetic Sequential Graceful Labeling For Arrow Graphs

Let G be a simple, finite, connected, undirected, non-trivial graph with 𝑝 vertices and 𝑞 edges. 𝑉(𝐺) be the vertex set and 𝐸(𝐺) be the edge set of 𝐺. Let 𝑓: 𝑉(𝐺) → {𝑎, 𝑎 + 𝑑, 𝑎 + 2𝑑, 𝑎 + 3𝑑, … , 𝑎 + 2𝑞𝑑} where a ≥ 0 and 𝑑 ≥ 1 is an injective function. If for each edge 𝑢𝑣 ∈ 𝐸(𝐺) , 𝑓 ∗ : 𝐸(𝐺) → {𝑑, 2𝑑, 3𝑑, 4𝑑, … , 𝑞𝑑} defined by 𝑓 ∗ (𝑢𝑣) = |𝑓(𝑢)− 𝑓(𝑣)| is a bijective function then the function 𝑓 is called arithmetic sequential graceful labeling. The graph with arithmetic sequential graceful labeling is called arithmetic sequential graceful graph. In this paper, we prove that one side arrow graphs 𝐴𝑅𝜂 2 , 𝐴𝑅𝜂 3 , 𝐴𝑅𝜂 5 and double-sided arrow graphs 𝐷(𝐴𝑅𝜂 2 ),𝐷(𝐴𝑅𝜂 3 ) are arithmetic sequential graceful graph

Complete characterization of bridge graphs with local antimagic chromatic number 2

An edge labeling of a connected graph $G = (V, E)$ is said to be local antimagic if it is a bijection $f\colon E \to\{1,\ldots ,|E|\}$ such that for any pair of adjacent vertices $x$ and $y$, $f^+(x)\not= f^+(y)$, where the induced vertex label $f^+(x)= \sum f(e)$, with $e$ ranging over all the edges incident to $x$. The local antimagic chromatic number of $G$, denoted by $\chi_{la}(G)$, is the minimum number of distinct induced vertex labels over all local antimagic labelings of $G$. In this paper, we characterize $s$-bridge graphs with local antimagic chromatic number 2.

Arithmetic Sequential Graceful Labeling of Complete Bipartite Graph with Pendant Edges

Arithmetic Sequential Graceful Labeling of Complete Bipartite Graph with Pendant Edges

L(3,2,1)-labeling of certain planar graphs

Given a graph G=(V,E) of maximum degree Δ, denoting by d(x,y) the distance in G between nodes x,y∈V, an L(3,2,1)-labeling of G is an assignment l from V to the set of non-negative integers such that |l(x)−l(y)|≥3 if x and y are adjacent, |l(x)−l(y)|≥2 if d(x,y)=2, and |l(x)−l(y)|≥1 if d(x,y)=3, for all x and y in V. The L(3,2,1)-number λ(G) is the smallest positive integer such that G admits an L(3,2,1)-labeling with labels from {0,1,…,λ(G)}.In this paper, the L(3,2,1)-number of certain planar graphs is determined, proving that it is linear in Δ, although the general upper bound for the L(3,2,1)-number of planar graphs is quadratic in Δ.

Asymmetric augmented paradigm-based graph neural architecture search

In most scenarios of graph-based tasks, graph neural networks (GNNs) are trained end-to-end with labeled samples. Labeling graph samples, a time-consuming and expert-dependent process, leads to huge costs. Graph data augmentations can provide a promising method to expand labeled samples cheaply. However, graph data augmentations will damage the capacity of GNNs to distinguish non-isomorphic graphs during the supervised graph representation learning process. How to utilize graph data augmentations to expand labeled samples while preserving the capacity of GNNs to distinguish non-isomorphic graphs is a challenging research problem. To address the above problem, we abstract a novel asymmetric augmented paradigm in this paper and theoretically prove that it offers a principled approach. The asymmetric augmented paradigm can preserve the capacity of GNNs to distinguish non-isomorphic graphs while utilizing augmented labeled samples to improve the generalization capacity of GNNs. To be specific, the asymmetric augmented paradigm will utilize similar yet distinct asymmetric weights to classify the real sample and augmented sample, respectively. To systemically explore the benefits of asymmetric augmented paradigm under different GNN architectures, rather than studying individual asymmetric augmented GNN (A2GNN) instance, we then develop an auto-search engine called Asymmetric Augmented Graph Neural Architecture Search (A2GNAS) to save human efforts. We empirically validate our asymmetric augmented paradigm on multiple graph classification benchmarks, and demonstrate that representative A2GNN instances automatically discovered by our A2GNAS method achieve state-of-the-art performance compared with competitive baselines. Our codes are available at: https://github.com/csubigdata-Organization/A2GNAS.

Odd-Even Congruence Labeling of Some Graphs

Introduction: Graph labeling is one of the fastest growing areas in the field of graph theory. A graph labeling is basically an assignment of integers to the nodes or edges or both, subject to certain conditions. Different types of graph labeling techniques have been developed by several authors. Objectives: The applications of labeling of graphs has diverse fields in the study of data base management, secret sharing schemes, physical cosmology, debug circuit design, X-ray, crystallography, astronomy, radar, broadcasting network, secret, coding theory and much more. The graph labeling techniques serve as useful mathematical models and solve various graph related problems.Methods: The strength of this research paper is based on Odd-even congruence labeling of different types of graphs. Assignment of natural numbers as labels for the edges and vertices of a graph. It is based on modular arithmetic property known as congruence graph labelling of a graph.For congruence graph labeling it entails the assignment of odd integers to vertices and even integers to edges with property of congruence graph labeling. Results: This labeling method has been identified on friendship graph, shell graph, generalized butterfly graph, fan graph, P_2+mK_1 graph

Linear Time Train Contraction Minor Labeling for Railway Line Capacity Analysis

AbstractWhen no more than one train is feasibly contained in the separation headway times of two other trains, a triangular gap problem-based method is used to compute the consumed capacity in linear time. This is a strong condition limiting its applicability, while real-world operations often feature mixed traffic of various speeds, creating more complex structures beyond the characterization of a triangular gap. In this paper, we attempt to investigate the multi-train gap scenario and provide a general solution to the railway timetable structure and capacity analysis problem. Given a timetable, we use an incidence graph, the so-called train contraction minor, for representing the consecutive train operations and show that a longest path for spanning the compressed timetable is any path in a graph induced by some topological subsequence of trains that connects the predefined vertices in the train contraction minor. By vector-valued vertex labeling of this minor, we acquire an efficient algorithm that computes the consumed capacity of the timetable. Our algorithm runs in O(mn) time, where m and n are the number of trains and stations, respectively, and is free from the limitation and outperforms the near-linear time policy iteration implementation in the max-plus system of train operations. A toy example and a real-world case study demonstrate the effectiveness and computational performance of the proposed method. The proposed method based on train contraction minor contributes to the railway operations community by promoting the efficient computation of railway line capacity into linear time.

SPORT: A Subgraph Perspective on Graph Classification with Label Noise

Graph neural networks (GNNs) have achieved great success recently on graph classification tasks using supervised end-to-end training. Unfortunately, extensive noisy graph labels could exist in the real world because of the complicated processes of manual graph data annotations, which may significantly degrade the performance of GNNs. Therefore, we investigate the problem of graph classification with label noise, which is demanding because of the complex graph representation learning issue and serious memorization of noisy samples. In this work, we present a novel approach called S ubgra p h Set Netw or k with Sample Selection and Consis t ency Learning (SPORT) for this problem. To release the overfitting of GNNs, SPORT proposes to characterize each graph as a set of subgraphs generated by certain predefined stratagems, which can be viewed as samples from its underlying semantic distribution in graph space. Then we develop an equivariant network to encode the subgraph set with the consideration of the symmetry group. To further release the influences of noisy examples, we leverage the predictions of subgraphs to measure the likelihood of a sample being clean or noisy, followed by effective label updating. In addition, we propose a joint loss to advance the model generalizability by introducing consistency regularization. Comprehensive experiments on a wide range of graph classification datasets demonstrate the effectiveness of our SPORT. Specifically, SPORT outperforms the most competing baseline by up to 6.4%.

A two-stage spatial prediction modeling approach based on graph neural networks and neural processes

Spatial prediction models hold significant application value in fields such as environmental science, economic development, and geological exploration. With advancements in deep learning technology, graph neural networks (GNNs) offer a powerful and scalable solution for spatial data modeling. In these models, each spatial location is represented as a vertex, with the explanatory variables at each point converted into vertex features, and the vertex label corresponding to the target value at that point. GNNs utilize this representation to predict vertex labels, thereby performing spatial prediction. However, the residuals of GNN predictions are found to still exhibit spatial autocorrelation, indicating that traditional GNNs only achieve suboptimal results in terms of capturing complex spatial relationships. In this paper, we propose a two-stage spatial prediction method called Location Embedded Graph Neural Networks-Residual Neural Processes (LEGNN-RNP) to address this challenge. In the first stage, we employ LEGNNs, a new framework that integrates location features into GNNs through an attention mechanism, enhancing the learning of complex spatial models by considering both spatial context and attribute features. In the second stage, we model the residuals from LEGNN predictions to further extract spatial patterns from the data. Specifically, we introduce RNP, a neural process-based approach to model the Gaussian Process (GP) distribution of residuals. The distribution parameters (i.e., mean and covariance) are parameterized by a neural network, where the mean estimates the residual and the variance quantifies the prediction uncertainty. Experiments on four datasets demonstrate that our proposed two-stage method achieves state-of-the-art results by effectively extracting spatial relationships and significantly improving spatial prediction accuracy.

A Family of Join Spider Graphs on Sum Divisor Cordial Labeling with Application

A graph G with p vertices and q edges is said to be a Sum Divisor Cordial labelling. If ????: ???? → 1,2, … ???? such that for each edge ???????????????? assign the label 1, if 2|(???? ???? + ????(????)| and 0 otherwise, satisfying the condition |e????(1)- e???? 0 | ≤ 1, where e????(1) is the number of edges labelled with 1 and e????(0) is the number of edges labelled with 0. A graph with a sum divisor cordial labeling graph. In this paper we study and we prove that for a class of Join Spider graphs with atmost one additional leg namely JSP(1m,2n), JSP(1m,2n,32), JSP(1m,2n,42), JSP(1m,2n,52). In this paper we identify an application using Affline Cipher for Cyber Security through encryption and decryption.