scholarly journals Super cyclic antimagic covering for some families of graphs

2021 ◽  
Vol 5 (1) ◽  
pp. 27-33
Author(s):  
Muhammad Numan ◽  
◽  
Saad Ihsan Butt ◽  
Amir Taimur ◽  
◽  
...  
Keyword(s):  
Coding Theory ◽  
Necessary Conditions ◽  
Graph Labeling ◽  
Computer Networking ◽  
Wheel Graph ◽  
Scheme Theory

Graph labeling plays an important role in different branches of sciences. It gives useable information in the study of radar, missile and rocket theory. In scheme theory, coding theory and computer networking graph labeling is widely employed. In the present paper, we find necessary conditions for the octagonal planner map and multiple wheel graph to be super cyclic antimagic cover and then discuss their super cyclic antimagic covering.

Total Irregularity Strengths of an Arbitrary Disjoint Union of (3,6)- Fullerenes

2020 ◽  
Vol 23 ◽  
Author(s):  
Ayesha Shabbir ◽  
Muhammad Faisal Nadeem ◽  
Mohammad Ovais ◽  
Faraha Ashraf ◽  
Sumiya Nasir
Keyword(s):  
Graph Theory ◽  
Lower Bound ◽  
Coding Theory ◽  
Disjoint Union ◽  
Fullerene Molecule ◽  
Graph Labeling ◽  
Fullerene Graph ◽  
Theoretical Concepts ◽  
Large Numbers ◽  
Fullerene Graphs

Aims and Objective: A fullerene graph is a mathematical model of a fullerene molecule. A fullerene molecule or simply a fullerene is a polyhedral molecule made entirely of carbon atoms other than graphite and diamond. Chemical graph theory is a combination of chemistry and graph theory where graph theoretical concepts used to study physical properties of mathematically modeled chemical compounds. Graph labeling is a vital area of graph theory which has application not only within mathematics but also in computer science, coding theory, medicine, communication networking, chemistry and in many other fields. For example, in chemistry vertex labeling is being used in the constitution of valence isomers and transition labeling to study chemical reaction networks. Method and Results: In terms of graphs vertices represent atoms while edges stand for bonds between atoms. By tvs (tes) we mean the least positive integer for which a graph has a vertex (edge) irregular total labeling such that no two vertices (edges) have same weights. A (3,6)-fullerene graph is a non-classical fullerene whose faces are triangles and hexagons. Here, we study the total vertex (edge) irregularity strength of an arbitrary disjoint union of (3,6)-fullerene graphs and providing their exact values. Conclusion: The lower bound for tvs (tes) depending on the number of vertices, minimum and maximum degree of a graph exists in literature while to get different weights one can use sufficiently large numbers, but it is of no interest. Here, by proving that the lower bound is the upper bound we close the case for (3,6)-fullerene graphs.

L(h,k)-Labeling of Intersection Graphs

2020 ◽  
pp. 135-170
Author(s):  
Sk. Amanathulla ◽  
Madhumangal Pal
Keyword(s):  
Coding Theory ◽  
Chordal Graph ◽  
Interval Graph ◽  
Graph Labeling ◽  
Frequency Assignment ◽  
Permutation Graph ◽  
Intersection Graphs ◽  
Circular Arc ◽  
Trapezoid Graph ◽  
Circular Arc Graphs

One important problem in graph theory is graph coloring or graph labeling. Labeling problem is a well-studied problem due to its wide applications, especially in frequency assignment in (mobile) communication system, coding theory, ray crystallography, radar, circuit design, etc. For two non-negative integers, labeling of a graph is a function from the node set to the set of non-negative integers such that if and if, where it represents the distance between the nodes. Intersection graph is a very important subclass of graph. Unit disc graph, chordal graph, interval graph, circular-arc graph, permutation graph, trapezoid graph, etc. are the important subclasses of intersection graphs. In this chapter, the authors discuss labeling for intersection graphs, specially for interval graphs, circular-arc graphs, permutation graphs, trapezoid graphs, etc., and have presented a lot of results for this problem.

Sum number of fans

2020 ◽  
pp. 2150058
Author(s):  
Tuga Mauritsius
Keyword(s):  
Communication Network ◽  
Lower Bounds ◽  
Secret Sharing ◽  
Coding Theory ◽  
Graph Labeling ◽  
Secret Sharing Schemes ◽  
Base Management System ◽  
Data Base Management System ◽  
Base Management ◽  
Positive Integers

Graph labeling deals with assigning labels to one or more elements of a graph. It has a wide variety of applications including: coding theory, communication network addressing, data base management system and secret sharing schemes to mention a view. A mapping [Formula: see text] is called a sum labeling of a graph [Formula: see text] if it is an injection from [Formula: see text] to a set of positive integers, such that [Formula: see text] if and only if there exists a vertex [Formula: see text] such that [Formula: see text]. In this case, [Formula: see text] is called a working vertex. In general, a graph [Formula: see text] will require some isolated vertices to be labeled in this way. The least possible number of such isolated vertices is called the sum number of [Formula: see text]; denoted by [Formula: see text]. A sum labeling of a graph [Formula: see text] is said to be optimum if it labels [Formula: see text] by using [Formula: see text] isolated vertices. In this paper, we investigate the lower bounds for the number of isolates required by an even fan and an odd fan, and then we construct optimum sum labelling for the graphs to prove: [Formula: see text]

scholarly journals Computing The Irregularity Strength of Planar Graphs

Mathematics ◽  
2018 ◽  
Vol 6 (9) ◽  
pp. 150 ◽  
Author(s):  
Hong Yang ◽  
Muhammad Siddiqui ◽  
Muhammad Ibrahim ◽  
Sarfraz Ahmad ◽  
Ali Ahmad
Keyword(s):  
Graph Theory ◽  
Circuit Design ◽  
Planar Graphs ◽  
Coding Theory ◽  
Vital Role ◽  
Graph Labeling ◽  
X Ray Crystallography ◽  
Radar Astronomy ◽  
Base Management ◽  
Irregularity Strength

The field of graph theory plays a vital role in various fields. One of the important areas in graph theory is graph labeling used in many applications such as coding theory, X-ray crystallography, radar, astronomy, circuit design, communication network addressing, and data base management. In this paper, we discuss the totally irregular total k labeling of three planar graphs. If such labeling exists for minimum value of a positive integer k, then this labeling is called totally irregular total k labeling and k is known as the total irregularity strength of a graph G. More preciously, we determine the exact value of the total irregularity strength of three planar graphs.

L(2,1,1)-Labeling of Circular-Arc Graphs

2021 ◽  
Vol 13 (2) ◽  
pp. 537-544
Author(s):  
S. Amanathulla ◽  
B. Bera ◽  
M. Pal
Keyword(s):  
Circuit Design ◽  
Coding Theory ◽  
Polynomial Time Algorithm ◽  
Time Algorithm ◽  
Graph Labeling ◽  
Frequency Assignment ◽  
Circular Arc ◽  
X Ray Crystallography ◽  
The Difference ◽  
Circular Arc Graphs

Graph labeling problem has been broadly studied in recent past for its wide applications, in mobile communication system for frequency assignment, radar, circuit design, X-ray crystallography, coding theory, etc. An L211-labeling  (L211L) of a graph G = (V, E) is a function γ : V → Z∗ such that |γ(u) − γ(v)| ≥ 2, if d(u, v) = 1 and |γ(u) − γ(v)| ≥ 1, if  d(u, v) = 1 or 2, where  Z∗  be the set of non-negative integers and d(u, v) represents the distance between the nodes u and v. The L211L numbers of a graph G, are denoted by λ2,1,1(G) which is the difference between largest and smallest labels used in L211L. In this article, for circular-arc graph (CAG) G we have proved that λ2,1,1(G) ≤ 6∆ − 4, where ∆ represents the degree of the graph. Beside this we have designed a polynomial time algorithm to label a CAG satisfying the conditions of L211L. The time complexity of the algorithm is O(n∆2), where n is the number of nodes of the graph G.

scholarly journals New classes of graphs with edge $ \; \delta- $ graceful labeling

2022 ◽  
Vol 7 (3) ◽  
pp. 3554-3589
Author(s):  
Mohamed R. Zeen El Deen ◽  
◽  
Ghada Elmahdy ◽  
Keyword(s):  
Positive Integer ◽  
Mathematical Models ◽  
Communication Networks ◽  
Data Security ◽  
Coding Theory ◽  
Graph Labeling ◽  
Graceful Labeling ◽  
Extensive Range ◽  
Fan Graph ◽  
Positive Integers

<abstract><p>Graph labeling is a source of valuable mathematical models for an extensive range of applications in technologies (communication networks, cryptography, astronomy, data security, various coding theory problems). An edge $ \; \delta - $ graceful labeling of a graph $ G $ with $ p\; $ vertices and $ q\; $ edges, for any positive integer $ \; \delta $, is a bijective $ \; f\; $ from the set of edge $ \; E(G)\; $ to the set of positive integers $ \; \{ \delta, \; 2 \delta, \; 3 \delta, \; \cdots\; , \; q\delta\; \} $ such that all the vertex labels $ \; f^{\ast} [V(G)] $, given by: $ f^{\ast}(u) = (\sum\nolimits_{uv \in E(G)} f(uv)\; )\; mod\; (\delta \; k) $, where $ k = max (p, q) $, are pairwise distinct. In this paper, we show the existence of an edge $ \; \delta- $ graceful labeling, for any positive integer $ \; \delta $, for the following graphs: the splitting graphs of the cycle, fan, and crown, the shadow graphs of the path, cycle, and fan graph, the middle graphs and the total graphs of the path, cycle, and crown. Finally, we display the existence of an edge $ \; \delta- $ graceful labeling, for the twig and snail graphs.</p></abstract>

scholarly journals Radio Reciprocal Membership Function on Cycle Related Graphs

2019 ◽  
Vol 8 (4) ◽  
pp. 626-628
Keyword(s):  
Membership Function ◽  
Graph Labeling ◽  
New Approach ◽  
Wheel Graph ◽  
Fan Graph ◽  
Radio Labeling

Radio labeling is graph labeling which deals with nodes of a graph. A new approach fuzzy radio reciprocal labeling proposed. Fuzzy radio reciprocal labeling deals with membership function [0,1] for every vertex and edge for making flexible which is stand for by -1 FRL .Fuzzy radio reciprocal labeling is determined for fan graph and wheel graph

Research on Information Process with a Computational Approach to Some Odd-Graceful Trees

2014 ◽  
Vol 1022 ◽  
pp. 207-210 ◽  
Author(s):  
Jian Min Xie ◽  
Bing Yao ◽  
Ming Yao ◽  
Xiang En Chen
Keyword(s):  
Communication Networks ◽  
Systems Engineering ◽  
Coding Theory ◽  
Computational Approach ◽  
Labeling Theory ◽  
Graph Labeling ◽  
Graph Colorings ◽  
Information Process ◽  
Engineering Theory ◽  
Olive Trees

Graph labeling theory has important applications in coding theory, communication networks, logistics and other aspects. In Operations Research or Systems Engineering Theory and Methods, one very often use graph colorings/labellings to divide large systems into subsystems. One can use colorings/labellings to distinguish vertices and edges between vertices in order to find fast algorithms to imitate some effective transmissions and communications in information networks. In this paper we present a computational approach to the odd-graceful labelings for some olive trees.

scholarly journals AI Uncertainty Based on Rademacher Complexity and Shannon Entropy

2021 ◽  
Vol 2 (1) ◽  
Keyword(s):  
Pattern Classification ◽  
Shannon Entropy ◽  
Error Rate ◽  
Channel Coding ◽  
Coding Theory ◽  
Communication Theory ◽  
Necessary Conditions ◽  
Classification Problems ◽  
Rademacher Complexity ◽  
Approach Zero

In this paper from communication channel coding perspective we are able to present both a theoretical and practical discussion of AI’s uncertainty, capacity and evolution for pattern classification based on the classical Rademacher complexity and Shannon entropy. First AI capacity is defined as in communication channels. It is shown qualitatively that the classical Rademacher complexity and Shannon rate in communication theory is closely related by their definitions. Secondly based on the Shannon mathematical theory on communication coding, we derive several sufficient and necessary conditions for an AI’s error rate approaching zero in classifications problems. A 1/2 criteria on Shannon entropy is derived in this paper so that error rate can approach zero or is zero for AI pattern classification problems. Last but not least, we show our analysis and theory by providing examples of AI pattern classifications with error rate approaching zero or being zero. Impact Statement: Error rate control of AI pattern classification is crucial in many lives related AI applications. AI uncertainty, capacity and evolution are investigated in this paper. Sufficient/necessary conditions for AI’s error rate approaching zero are derived based on Shannon’s communication coding theory. Zero error rate and zero error rate approaching AI design methodology for pattern classifications are illustrated using Shannon’s coding theory. Our method shows how to control the error rate of AI, how to measure the capacity of AI and how to evolve AI into higher levels. Index Terms: Rademacher Complexity, Shannon Theory, Shannon Entropy, Vapnik-Cheronenkis (VC) dimension.

Coding Theory

2004 ◽  
Author(s):  
San Ling ◽  
Chaoping Xing
Keyword(s):  
Coding Theory
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