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.

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.

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 Intuitionistic Fuzzy Planar Graphs

2014 ◽  
Vol 2014 ◽  
pp. 1-9 ◽  
Author(s):  
Noura Alshehri ◽  
Muhammad Akram
Keyword(s):  
Graph Theory ◽  
Fuzzy Sets ◽  
Fuzzy Set ◽  
Planar Graphs ◽  
Intuitionistic Fuzzy Set ◽  
Intuitionistic Fuzzy Sets ◽  
Dual Graphs ◽  
Intuitionistic Fuzzy

Graph theory has numerous applications in modern sciences and technology. Atanassov introduced the concept of intuitionistic fuzzy sets as a generalization of fuzzy sets. Intuitionistic fuzzy set has shown advantages in handling vagueness and uncertainty compared to fuzzy set. In this paper, we apply the concept of intuitionistic fuzzy sets to multigraphs, planar graphs, and dual graphs. We introduce the notions of intuitionistic fuzzy multigraphs, intuitionistic fuzzy planar graphs, and intuitionistic fuzzy dual graphs and investigate some of their interesting properties. We also study isomorphism between intuitionistic fuzzy planar graphs.

Using Apps to Visualize Graph Theory

2015 ◽  
Vol 108 (8) ◽  
pp. 626-631 ◽  
Author(s):  
Anne Quinn
Keyword(s):  
Graph Theory ◽  
Planar Graphs ◽  
Dynamic Graph ◽  
Platonic Solids ◽  
Matrix Representations

An inexpensive dynamic graph theory app can be used for matrix representations, planar graphs, Platonic solids, and more.

Using Dynamic Geometry Software to Teach Graph Theory: Isomorphic, Bipartite, and Planar Graphs

1997 ◽  
Vol 90 (4) ◽  
pp. 328-332
Author(s):  
Anne Larson Quinn
Keyword(s):  
Graph Theory ◽  
Planar Graphs ◽  
Software Package ◽  
Dynamic Geometry ◽  
Bipartite Graphs ◽  
Dynamic Geometry Software ◽  
Discrete Mathematics ◽  
Geometer's Sketchpad ◽  
Mathematics Courses ◽  
Mathematics Course

I have always used concrete marupulatives, such as marshmallows and toothpicks, to create models for my geometry and discrete-mathematics courses. These models have come in handy when discussing volume, introducing the 4-cube, or illustrating isomorphic or bipartite graphs. However, after discovering what a dynamic geometry–software package could do for geometry teaching, which has been well documented by research (e.g., Battista and Clements [1995]), I realized that this type of technology also had much to offer for teaching graph theory in my discrete-mathematics course. Although this article discusses The Geometer's Sketchpad 3 (Jackiw 1995), any software that can draw, label, and drag figures can be substituted for Sketchpad.

A Characterization of Almost-Planar Graphs

1996 ◽  
Vol 5 (3) ◽  
pp. 227-245 ◽  
Author(s):  
Bradley S. Gubser
Keyword(s):  
Graph Theory ◽  
Planar Graph ◽  
Planar Graphs ◽  
Explicit Description ◽  
Famous Result ◽  
Minor Criterion ◽  
Excluded Minor

Kuratowski's Theorem, perhaps the most famous result in graph theory, states that K5 and K3,3 are the only non-planar graphs for which both G\e, the deletion of the edge e, and G/e, the contraction of the edge e, are planar for all edges e of G. We characterize the almost-planar graphs, those non-planar graphs for which G\e or G/e is planar for all edges e of G. This paper gives two characterizations of the almost-planar graphs: an explicit description of the structure of almost-planar graphs; and an excluded minor criterion. We also give a best possible bound on the number of edges of an almost-planar graph.

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.

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