Edge Labels Research Articles

Graceful game on some graph classes

A graceful labeling of a graph G with m edges consists in labeling the vertices of G with distinct integers from 0 to m such that each edge is uniquely identified by the absolute difference of the labels of its endpoints. In this work, we study the graceful labeling problem in the context of maker-breaker graph games. The Graceful Game was introduced by Tuza, in 2017, as a two-players game on a connected graph in which the players, Alice and Bob, take moves labeling the vertices with distinct integers from 0 to m. Players are constrained to use only legal labelings (moves), that is, after a move, all edge labels are distinct. Alice’s goal is to obtain a graceful labeling for the graph, as Bob’s goal is to prevent it from happening. In this work, we study winning strategies for Alice and Bob in graph classes: paths, complete graphs, cycles, complete bipartite graphs, caterpillars, trees, gear graphs, web graphs, prisms, hypercubes, 2-powers of paths, wheels and fan graphs.

Holomorphism and Edge Labeling: An Inner Study of Latin Squares Associated with Antiautomorphic Inverse Property Moufang Quasigroups with Applications

Holomorphism and Edge Labeling: An Inner Study of Latin Squares Associated with Antiautomorphic Inverse Property Moufang Quasigroups with Applications

Domain-Adaptive Graph Attention-Supervised Network for Cross-Network Edge Classification.

Graph neural networks (GNNs) have shown great ability in modeling graphs; however, their performance would significantly degrade when there are noisy edges connecting nodes from different classes. To alleviate negative effect of noisy edges on neighborhood aggregation, some recent GNNs propose to predict the label agreement between node pairs within a single network. However, predicting the label agreement of edges across different networks has not been investigated yet. Our work makes the pioneering attempt to study a novel problem of cross-network homophilous and heterophilous edge classification (CNHHEC) and proposes a novel domain-adaptive graph attention-supervised network (DGASN) to effectively tackle the CNHHEC problem. First, DGASN adopts multihead graph attention network (GAT) as the GNN encoder, which jointly trains node embeddings and edge embeddings via the node classification and edge classification losses. As a result, label-discriminative embeddings can be obtained to distinguish homophilous edges from heterophilous edges. In addition, DGASN applies direct supervision on graph attention learning based on the observed edge labels from the source network, thus lowering the negative effects of heterophilous edges while enlarging the positive effects of homophilous edges during neighborhood aggregation. To facilitate knowledge transfer across networks, DGASN employs adversarial domain adaptation to mitigate domain divergence. Extensive experiments on real-world benchmark datasets demonstrate that the proposed DGASN achieves the state-of-the-art performance in CNHHEC.

Perfect Matching and Zero-Sum 3-Magic Labeling

A mapping l:E(G)→A, where A is an abelian group written additively, is called an edge labeling of the graph G. For every positive integer h⩾2, a graph G is said to be zero-sum h-magic if there is an edge labeling l from E(G) to Zh∖{0} such that s(v)=∑uv∈E(G)l(uv)=0 for every vertex v∈V(G). In 2014, Saieed Akbari, Farhad Rahmati and Sanaz Zare proved that if r (r≠5) is odd and G is a 2-edge connected r-regular graph, G admits a zero-sum 3-magic labeling, and they also conjectured that every 5-regular graph admits a zero-sum 3-magic. In this paper, we first prove that every 5-regular graph with every edge contained in a triangle must have a perfect matching, and then we denote the edge set of the perfect maching by EM, and we make a labeling l:E(EM)→2, and E(E(G)–EM)→1. Thus we can easily see this labeling is a zero-sum 3-magic, confirming the above conjecture with a moderate condition.

ABSOLUTE MEAN GRACEFUL LABELING IN THE CONTEXT OF m-SPLITTING AND DEGREE SPLITTING GRAPHS

A graph G with q edges is said to be absolute mean graceful if there is a one-to-one function f from V (G) to the set {0,±1,±2,±3,...,±q} such that when each edge xy is assigned the label |f(x)−f(y)| 2 , then the resulting edge labels are distinct. In this paper, the absolute mean graceful labeling of m-splitting and degree splitting graphs of some graphs are investigated.

EDGE IRREGULAR REFLEXIVE LABELING ON MONGOLIAN TENT GRAPH (M_(m,3)) AND DOUBLE QUADRILATERAL SNAKE GRAPH

Let G be an undirected, connected, and simple graph with edges set E(G)and vertex set V(G). An edge irregular reflexive k-labeling f is one in which the label for each edge is an integer number {1,2,…, k_e} and the label for each vertex is an even integer number {0,2,4,…,2k_v}, k = max{ k_e,2k_v}. This type of labeling results in distinct weights for each edge. The weight of an edge xy in a graph G with labeling f, indicated by wt (xy), is the total of the labels on the vertex that are incident to the edge as well as the edge label. The minimum value k of the largest label in the graph G is referred to as res (G), which stands for the reflexive edge strength of the graph G. The topic of edge irregular reflexive k-labeling for mongolian tent graph (M_(m,n)) and double quadrilateral snake graph (D(Q_n )) will be discussed in this paper. The res (M_(m,n)),m≥2,n=3 has been obtained that is ⌈(5m-1)/3⌉ for 5m-1≢2,3 (mod 6) and ⌈(5m-1)/3⌉+1 for 5m-1≡2,3 (mod 6). Also the res (D(Q_n )),n≥2 has been obtained that is ⌈(7n-7)/3⌉ for 7n-7≢2,3 (mod 6) and ⌈(5m-1)/3⌉+1 for 7n-7≡2,3 (mod 6).

EDGE IRREGULAR REFLEXIVE LABELING ON ALTERNATE TRIANGULAR SNAKE AND DOUBLE ALTERNATE QUADRILATERAL SNAKE

Let G in this paper be a connected and simple graph with set V(G) which is called a vertex and E(G) which is called an edge. The edge irregular reflexive k-labeling f on G consist of integers {1,2,3,...,k_e} as edge labels and even integers {0,2,4,...,2k_v} as the label of vertices, k=max{k_e,2k_v}, all edge weights are different. The weight of an edge xy in G represented by wt(xy) is defined as wt(xy)= f (x)+ f (xy)+ f (y). The smallest k of graph G has an edge irregular reflexive k-labeling is called the reflexive edge strength, symbolized by res (G). In article, we discuss about edge irregular reflexive k-labeling of alternate triangular snake A(T_n ) and the double alternate quadrilateral snake DA(Q_n ). In this paper, the res of alternate triangular snake A(T_n ) , n≥3 has been obtained. That is ⌈(2n-1)/3⌉ for n even,2n-1≢2,3 (mod 6),⌈(2n-1)/3⌉+1 for n even,2n-1=2,3 (mod 6),⌈(2n-2)/3⌉ for n odd,2n-2≢2,3 (mod 6), and ⌈(2n-2)/3⌉+1 for n odd,2n-2=2,3 (mod 6). Then, the reflexive edge strength of double alternate quadrilateral snake DA (Q_n) ⌈ (4n-1 )/3⌉for n even, 4n - 1 ≠2,3 (mod 6), ⌈ (4n-1 )/3⌉+1 for n even, 4n - 1 = 2,3 (mod 6), ⌈ (4n-4)/3⌉ for n odd, 4n - 4 ≠2,3 (mod 6), and ⌈ (4n-4 )/3⌉+1 for n odd, 4n - 4 = 2,3 (mod 6).

Conservative trees

The conservative number of a graph G is the minimum M such that G admits an orientation and a labeling of its edges with distinct numbers in {1,2,…,M} so that at each vertex of degree at least three, the sum of the labels of the incoming edges minus the sum of the labels of the outgoing edges is zero. A graph is conservative if its conservative number and its size are equal.In this work we determine the conservative number of several classes of trees and study the relation between conservative trees and graceful cycles with chords. We also show that for a given size of its base cycle, shell-type graphs with the maximum number of consecutive 4-faces are graceful.

On Leech labelings of graphs and some related concepts

Let G=(V,E) be a graph and let f:E→{1,2,3,…} be an edge labeling of G. The path weight of a path P in G is the sum of the labels of the edges of P and is denoted by w(P). The path number of G, tp(G) is the total number of paths in a graph G. If the set of all path weights S in G with respect to the labeling f is {1,2,3,…,tp(G)}, then f is called a Leech labeling of G. A graph which admits a Leech labeling is called a Leech graph. Leech index is a parameter which evaluates how close a graph is towards being Leech. In this paper, the path number of the wheel graph Wn is obtained. We also determine a bound for the Leech index of Wn and a subclass of unicyclic graphs. A python program that gives all possible Leech labelings of a cycle Cn for n≥3, if it exists, is also provided.

Anti Skolem Mean Labeling of Quardrilateral Snake Related Graphs

A graph with p vertices and q edges where is said to be an anti skolem mean graph if it is possible to label the vertices with distinct labels from in such a way that each edge e=uv is labelled with then the resulting edge labels are distinct labels from the set . In this case f is called an Anti Skolem Mean labelling of G.

CoCo-trie: Data-aware compression and indexing of strings

We address the problem of compressing and indexing a sorted dictionary of strings to support efficient lookups and more sophisticated operations, such as prefix, predecessor, and range searches. This problem occurs as a key task in a plethora of applications, and thus it has been deeply investigated in the literature since the introduction of tries in the ’60s.We introduce a new data structure, called the COmpressed COllapsed Trie (CoCo-trie), that hinges on a pool of techniques to compress subtries (of arbitrary depth) into succinctly-encoded and efficiently-searchable trie macro-nodes with a possibly large fan-out. Then, we observe that the choice of the subtries to compress depends on the trie structure and its edge labels. Hence, we develop a data-aware optimisation approach that selects the best subtries to compress via the above pool of succinct encodings, with the overall goal of minimising the total space occupancy and still achieving efficient query time. We also investigate some variants of this approach that induce interesting space–time trade-offs in the CoCo-trie design.Our experimental evaluation on six diverse and large datasets (representing URLs, XML data, DNA and protein sequences, database records, and search-engine dictionaries) shows that the space–time performance of well-established and highly-engineered data structures solving this problem is very input-sensitive. Conversely, our CoCo-trie provides a robust and uniform improvement over all competitors for half of the datasets, and it results on the Pareto space–time frontier for the others, thus offering new competitive trade-offs.

Rough Set Techniques in Wireless Sensor Networks using Membership Function and Rough Labelling Graphs for Energy Aware Routing

Rough set theory and rough graphs are employed for data analysis. Rough graphs utilize approximations, and this paper presents a method for implementing rough graphs using rough membership functions and graph labeling for data structures and reduction. This paper proposes a rough graph labeling method, termed rough -labeling similarity graph, that utilizes a similarity measure for vertex and edge labeling. This method aims to minimize boundary regions in rough graphs and is applicable to wireless sensor networks (WSNs). In WSN, the proposed algorithms such as PSO-LSTM, COA-LSTM, LOA-LSTM integrates with rough set theory based rough -labeling for boundary region identification. The proposed method incorporates the membership functions encapsulates the rough labeling graphs for cluster boundary region identification for WSN. The cluster boundary for WSN is based on rough set membership and rough labelling is termed as rough set membership boundary region (RMB). RMB in WSN and implementation of proposed routing algorithm such as LOA-LSTM provides high throughput and energy saving when compared to existing cluster boundary structure methods such as Voronoi, Spectrum and Chain.

The linear framework II: using graph theory to analyse the transient regime of Markov processes.

The linear framework uses finite, directed graphs with labelled edges to model biomolecular systems. Graph vertices represent chemical species or molecular states, edges represent reactions or transitions and edge labels represent rates that also describe how the system is interacting with its environment. The present paper is a sequel to a recent review of the framework that focussed on how graph-theoretic methods give insight into steady states as rational algebraic functions of the edge labels. Here, we focus on the transient regime for systems that correspond to continuous-time Markov processes. In this case, the graph specifies the infinitesimal generator of the process. We show how the moments of the first-passage time distribution, and related quantities, such as splitting probabilities and conditional first-passage times, can also be expressed as rational algebraic functions of the labels. This capability is timely, as new experimental methods are finally giving access to the transient dynamic regime and revealing the computations and information processing that occur before a steady state is reached. We illustrate the concepts, methods and formulas through examples and show how the results may be used to illuminate previous findings in the literature.

Finding Minimum Connected Subgraphs With Ontology Exploration on Large RDF Data

In this paper, we study the following problem: given a knowledge graph (KG) and a set of input vertices (representing concepts or entities) and edge labels, we aim to find the smallest connected subgraphs containing all of the inputs. This problem plays a key role in KG-based search engines and natural language question answering systems, and it is a natural extension of the Steiner tree problem, which is known to be NP-hard. We present RECON, a system for finding approximate answers. RECON aims at achieving high accuracy with instantaneous response (i.e., sub-second/millisecond delay) over KGs with hundreds of millions edges without resorting to expensive computational resources. Furthermore, when no answer exists due to disconnection between concepts and entities, RECON refines the input to a semantically similar one based on the ontology, and attempts to find answers with respect to the refined input. We conduct a comprehensive experimental evaluation of RECON. In particular we compare it with five existing approaches for finding approximate Steiner trees. Our experiments on four large real and synthetic KGs show that RECON significantly outperforms its competitors and incurs a much smaller memory footprint.

Elastic founder graphs improved and enhanced

Indexing labeled graphs for pattern matching is a central challenge of pangenomics. Equi et al. (2022) [14] developed the Elastic Founder Graph (EFG) representing an alignment of m sequences of length n, drawn from alphabet Σ plus the special gap character: the paths spell the original sequences or their recombination. By enforcing the semi-repeat-free property, the EFG admits a polynomial-space index for linear-time pattern matching, breaking through the conditional lower bounds on indexing labeled graphs (Equi et al. [13]). In this work, we improve the space of the EFG index answering pattern matching queries in linear time, from linear in the length of all strings spelled by three consecutive node labels, to linear in the size of the edge labels. Then, we develop linear-time construction algorithms optimizing for different metrics: we improve the existing linearithmic construction algorithms to O(mn), by solving the novel exclusive ancestor set problem on trees; we propose, for the simplified gapless setting, an O(mn)-time solution minimizing the maximum block height, that we generalize by substituting block height with prefix-aware height. Finally, to show the versatility of the framework, we develop a BWT-based EFG index and study how to encode and perform document listing queries on a set of paths of the graphs, reporting which paths present a given pattern as a substring. We propose the EFG framework as an improved and enhanced version of the framework for the gapless setting, along with construction methods that are valid in any setting concerned with the segmentation of aligned sequences.

Group-supermagic labeling of Cartesian products of two even cycles

A Γ-supermagic labeling of a graph G=(V,E) with |E|=q is a bijection from E to an Abelian group Γ of order q such that the sum of labels of all incident edges of every vertex x∈V is equal to the same element μ∈Γ. An existence of a Γ-supermagic labeling of Cartesian product of two cycles Cm□Cn for every m,n≥3 by Z2mn and for any two odd cycles Cm□Cn for any m,n≥3 and any Abelian group Γ of order 2mn was proved recently. In this paper we completely solve the case of m,n both even and present a labeling method for all Abelian groups of order 2mn, where m,n are even and greater than two.

On local antimagic chromatic numbers of circulant graphs join with null graphs or cycles

An edge labeling of a graph G = (V,E) is said to be local antimagic if there is a bijection f : E → {1,..., |E|} such that for any pair of adjacent vertices x and y, f +(x) ≠ f +(y), where the induced vertex label is f +(x) = 𝜮 f(e), with e ranging over all the edges incident to x. The local antimagic chromatic number of G, denoted by χla(G), is the minimum number of distinct induced vertex labels over all local antimagic labelings of G. For a bipartite circulant graph G, it is known that χ(G)=2 but χla(G) ≥ 3. Moreover, χla(Cn ∨ K1)=3 (respectively 4) if n is even (respectively odd). Let G be a graph of order m ≥ 3. In [Affirmative solutions on local antimagic chromatic number, Graphs Combin., 36 (2020), 1337—1354], the authors proved that if m ≡ n (mod 2) with χla(G) = χ(G), m>n ≥ 4 and m ≥ n2/2, then χla(G∨On) = χla(G)+1. In this paper, we show that the conditions can be omitted in obtaining χla(G∨H) for some circulant graph G, and H is a null graph or a cycle. The local antimagic chromatic number of certain wheel related graphs are also obtained.

Analytic odd mean labeling of union and identification of some graphs

A graph G is analytic odd mean if there exist an injective function f : V → {0, 1, 3, . . . , 2q − 1} with an induced edge labeling f∗ : E → Z such that for each edge uv with f(u) < f(v), is injective. Clearly the values of f∗ are odd. We say that f is an analytic odd mean labeling of G. In this paper, we show that the union and identification of some graphs admit analytic odd mean labeling by using the operation of joining of two graphs by an edge.

Dense subgraphs induced by edge labels

Finding densely connected groups of nodes in networks is a widely-used tool for analysis in graph mining. A popular choice for finding such groups is to find subgraphs with a high average degree. While useful, interpreting such subgraphs may be difficult. On the other hand, many real-world networks have additional information, and we are specifically interested in networks with labels on edges. In this paper, we study finding sets of labels that induce dense subgraphs. We consider two notions of density: average degree and the number of edges minus the number of nodes weighted by a parameter α\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\alpha$$\\end{document}. There are many ways to induce a subgraph from a set of labels, and we study two cases: First, we study conjunctive-induced dense subgraphs, where the subgraph edges need to have all labels. Secondly, we study disjunctive-induced dense subgraphs, where the subgraph edges need to have at least one label. We show that both problems are NP-hard. Because of the hardness, we resort to greedy heuristics. We show that we can implement the greedy search efficiently: the respective running times for finding conjunctive-induced and disjunctive-induced dense subgraphs are in Oplogk\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathcal {O} \\mathopen {}\\left( p \\log k\\right)$$\\end{document} and Oplog2k\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathcal {O} \\mathopen {}\\left( p \\log ^2 k\\right)$$\\end{document}, where p is the number of edge-label pairs and k is the number of labels. Our experimental evaluation demonstrates that we can find the ground truth in synthetic graphs and that we can find interpretable subgraphs from real-world networks.

SUM CORDIAL LABELING OF CORONA PRODUCT GRAPH OF PATH TO PATH

In this paper the researcher studies on the labeling of Corona Product Graph of Path to Path by admitting the certain condition of Sum cordial labeling. A sum cordial labeling of a graph is a binary labeling with vertex set V, with edge labeling f:E(G) {0,1},defined by f(uv)=(f(u)+f(v))(mod2) if |vf(0)-vf(1)|≤1and |ef(0)-ef(1)|≤1. A graph G is Sum cordial if it admits Sum cordial labeling.