The notes form the base text for the course mat62756 graph theory. Some new results on prime graphs scientific research publishing. We also discuss prime labeling in the context of graph operations namely duplication in k n e. Graph labeling is an important area of research in graph theory.
We investigate prime labeling for some graphs resulted by identifying any two vertices of some graphs. A coprime labeling of a simple graph of order n is a labeling in which adjacent vertices are given relatively prime labels, and a graph is prime if the labels used can be taken to be the rst n positive integers. Square difference prime labeling more results on path related graphs chapter 12 advances in mathematics and computer science vol. A coprime labeling of a simple graph of order n is a labeling in which adjacent vertices are given relatively prime labels, and a graph is prime if the labels used can be taken to be the. Reinhard diestel graph theory electronic edition 2000 c springerverlag new york 1997, 2000 this is an electronic version of the second 2000 edition of the above springer book, from their series graduate texts in mathematics, vol. Some of the major themes in graph theory are shown in figure 3. In the mathematical discipline of graph theory, a graph labeling is the assignment of labels, traditionally represented by integers, to edges andor vertices of a graph. Here, we investigate and prove that new results for triangular book admits prime labeling when is even and odd. We also show that the square graph of bn,n is a prime cordial graph while middle graph of pn is a prime cordial graph for n. An outline of the results contained in all the chapters is given in the introduction. Aug 24, 2011 in the first and second parts of my series on graph theory i defined graphs in the abstract, mathematical sense and connected them to matrices. We also introduce the concept of strongly prime graph and prove that the graphs cn, pn, and k1,n are strongly prime graphs. In the present work we investigate some classes of graphs which admit prime. Problems and conjectures in graceful labelings, harmonious labelings, prime labelings, additive labelings, and zonal labelings are introduced with fundamentals, examples, and illustrations.
Umbrella graph, p nqs n graph, c nq sn graphs are square difference graphs. All graphs in this paper are finite and undirected. Although the conjecture that all trees are graceful has been the focus of many of these papers, this conjecture is still unproved. Recall that a graph is a collection of vertices or nodes and edges between them. Square difference labeling, square difference graph. A prime labeling of a graph of order is an injective function such that for every pair of adjacent vertices and. A graph g v, e with n vertices is said to admit prime labeling if its vertices can be labeled with distinct positive integers not exceeding n such that the labels of each pair of adjacent. A graph g is a prime distance graph if its vertices can be labeled with distinct integers in such a way that for any two adjacent vertices, the absolute difference of their labels is a prime number. In this paper we have proved that some classes of graphs such as the flower pot, coconut tree, umbrella graph, shell graph, carona of a hell graph,s carona of a wheel graph, carona of a ear graphg, butterfly graph, two copies of cycle. Introduction all graphs in this paper are simple finite undirected and nontrivial graph gv, e with vertex set v and the edge set e. It is known that cycles and bipartite graphs are prime distance graphs. Here we extend the idea of prime labeling to the gaussian integers, which are the complex numbers whose real and imaginary parts are both integers. We show that the graphs kayak paddle kpk,m,l, book graph. Moreover we prove that wn is a strongly prime graph for every even integer n.
For the remainer of this paper whenever refering to a graph we will be refering to an edge labeled graph. A graph with vertex set is said to have a prime labeling if its vertices are labeled with distinct integers such that for each edge the labels assigned to and are relatively prime. In this part well see a real application of this connection. They also show that a 2regular graph with at least two odd cycles has no vertex prime labeling.
Introduction if the vertices of the graph are assigned values subject to certain conditions is known as graph labeling and have been motivated by their utility to various applied fields and their intrinsic mathematical interest. One of the important areas in graph theory is graph labeling used in many applications like coding theory, xray crystallography, radar, astronomy, circuit design, communication network addressing, data base management. Prime vertex labelings of several families of graphs. The length of the lines and position of the points do not matter. All unicyclic graphs have a prime vertex labeling seoud and youssef 5. Neighbourhood prime labeling on some graphs semantic scholar. The square sum labeling and square sum graphs are 1 are defined and discussed by v. As a research area, graph theory is still relatively young, but it is maturing rapidly with many deep results having been discovered over the last couple of decades. Formally, given a graph g v, e, a vertex labeling is a function of v to a set of labels.
During the past thirty years, over 200 papers on this topics have been appeared in journals. Z, in other words it is a labeling of all edges by integers. By a prime labeling, we mean a way to label the n vertices with the integers 1 to n such that each pair of adjacent vertices is relatively prime. It took a hundred years before the second important contribution of kirchhoff 9 had been made for the analysis of. Neighborhoodprime labelings of trees and other classes of graphs. This outstanding book cannot be substituted with any other book on the present textbook market. In this paper we investigate prime labelling of some new graphs. Pdf some vertex prime graphs and a new type of graph labeling. On neighbourhood prime labeling of certain classes of graphs. Pdf a graph g v, e with n vertices is said to admit prime labeling if its vertices. Most of these topics have been discussed in text books.
We also discuss prime labeling in the context of some graph operations namely fusion, duplication and vertex switching in cycle cn. In a weighted graph, the weight of a path is the sum of the weights of the edges traversed. Graph is a mathematical representation of a network and it describes the relationship between lines and points. Free graph theory books download ebooks online textbooks. A labeling or numbering of a graph g with q edges is an assignment of labels to the vertices of g that induces for each edge uv a labeling depending on the. For graph theoretic terminology, we refer to harary 2. In this paper we derive certain general results concerning prime distance labeling. It contains three types of vertices, an apex of degree 2n, n vertices of degree 4 and n vertices of degree 2. Buy graph theory book online at best prices in india on. A dynamic survey on graph labeling is regularly updated by gallian 1 and it is published in electronic journal of combinatorics.
This is a serious book about the heart of graph theory. Labeling of a graph g is an assignment of integers either to the vertices or edges or both subject to certain conditions 2,3. A graph which satisfies the conditions of prime labeling and harmonious labeling is called a prime harmonious labeling. Square difference prime labeling of some planar graphs. Then a prime labeling ensures that when there is a change of guard at one station, there is not simultaneously a change of guard at all the adjacent stations, until the lcm of the labels in the neighborhood is reached. I was thinking of any applications like scheduling etc. Square difference labeling, square difference graph, cycle cactus. Abstracta graph, with n vertices is said to admit prime labeling if its vertices can be labeled with distinct positive integers not exceed n such that the label of each pair of adjacent vertices are relatively prime. An edgegraceful labelling on a simple graph without loops or multiple edges on p vertices and q edges is a labelling of the edges by distinct integers in 1, q such that the labelling on the vertices induced by labelling a vertex with the sum of the incident edges taken modulo p assigns all values from 0 to p. If you dont want to be overwhelmed by doug wests, etc. A graph in this context is made up of vertices also called nodes or points which are connected by edges also called links or lines.
In the intervening years dozens of graph labelings techniques have been studied in over 1700 papers. Under the umbrella of social networks are many different types of graphs. We prove that the graphs such as flower graph f, the splitting graph of star, the bistar, the friendship graph the graph sfn,1 are prime. In this paper we have investigated that the grotzsch graph is a prime graph and graph operations namely, fusion, duplication and switching on grotzsch graph admits prime labeling. In general, all the graphs are not prime, it is very interesting to investigate graph families which admit prime labelling. The labeling of the vertices respectively edges is injective if distinct vertices respectively edges have distinct labels. A graph with vertex set v is said to have a prime labeling if its vertices are labeled with distinct integer 1,2,3 such that for edge the labels assigned to and are relatively prime. Applications of graph labeling in communication networks. It has every chance of becoming the standard textbook for graph theory. A difference labeling of g is an injection f from v to the set of non. We introduce a new type of graph labeling called as lcordial labeling and show that k 1,n,path p n, c n,sc 3.
Let g be an undirected graph without loops or double connections between vertices. In this paper we investigate prime labeling for k n e where e is an edge. Nagarajan 1 investigate neighbourhood prime labeling for some special graphs like friendship graph, gear, ladder, triangular book and coconut tree. They proved that the cycle, complete graph kn, the cycle cactus, ladder and complete lattice grids are square. A graph labeling is an assignment of integers to the vertices or edges, or both, subject to certain conditions. A prime cordial labeling of a graph with the vertex set is a bijection such that each edge is assigned the label 1 if and 0 if. The helm graph is a labeled graph with common weight decomposition. In this paper we investigate prime labeling for some fan related graphs. In this paper we investigate some new families of vertex prime graphs. Some vertex prime graphs and a new type of graph labeling. This book could give me the necessary basis to understand the terminology and principles, as well as methods to provide proof, to use the theory for an unusual application i am interested in.
Prime labeling for some cycle related graphs semantic scholar. It covers all the topics required for an advanced undergrad course or a graduate level graph theory course for math, engineering, operations research or. Although beyond the scope of this leisurely introduction to knot theory, one of the most successfull and interesting ways to tell knots apart is through the various knot polynomials, of which there is an incredible variety. In this paper we have proved that some classes of graphs such as the flower pot, coconut tree, umbrella graph, shell graph. Graph obtained by identifying a rim vertex of w 8 with an end vertex of p 6 and its prime labeling. P2 with the value of 1 assigned to the top left vertex of the graph is not a prime labeling. The aim of journal of graph labeling is to bring together original and significant research articles in different areas of graph labeling and graph coloring. In this paper we prove that the split graphs of k1,n and bn,n are prime cordial graphs. This work is a nice combination of graph theory and elementary number theory. As all the graphs are not prime graph it is very interesting to investigate graph or graph families which admit prime labeling.
A graph which admits a prime cordial labeling is called a prime cordial graph. A graph with such a labeling is an edge labeled graph. A graph g with n vertices is said to admit prime labeling if its vertices can be labeled with distinct positive integers not exceeding n such that the labels of each pair of adjacent vertices are relatively prime. Graph theory is a very popular area of discrete mathematics with not only numerous theoretical developments, but also countless applications to practical problems. What are primes in graphs and how many of them have a given length. A common theme in graph labeling papers is to build up graphs that have.
Prime labeling of grotzch graph mathematics journal, ijmtt. The dots are called nodes or vertices and the lines are called edges. In mathematics, graph theory is the study of graphs, which are mathematical structures used to model pairwise relations between objects. The symbol vg and eg denotes the vertex set and edge set of a graph g. Cahit3 introduced cordial labeling of graphs and derived various results on cordial graphs. A graph which admits prime cordial labeling is called prime cordial graph. Yellen, graph theory and its applications, crc press, boca raton, 1999. Theory and applications labeled graphs are becoming an increasingly useful family of mathematical models for a broad range of applications. The concept of prime cordial labeling was introduced by sundaram 5 et al. A graph that admits a prime labelling is called a prime graph.
Gallians survey paper on graph labelings claims one exists and cites fu and huangs on prime labelling, which in turn cites an unpublished paper for this result. A graph labeling is an assignment of integers to the vertices or edges or both subject to certain conditions. B, where b is the book with triangular and rectangle pages, g mn. V g 1,2,p is called a prime labeling if for each edge euv, gcd f u, f v1. Bn,m, and k neighborhood prime labeling of paths and some special graphs in 5.
The field of graph theory plays vital role in various fields. I have a mathematical background, but am not an expert in graph theory. In this paper the prime labeling of certain classes of graphs are discussed. A graph which admits a prime harmonious labeling is called a prime harmonious graph and it is denoted by p h. In this paper an analysis is made on union of graphs are prime cordial labeling. The graph admitting vertex prime labeling is called as verterx prime graph. The notion of prime labeling was originated by entringer and was discussed in a. Online shopping for graph theory from a great selection at books store. A graph g which admits prime labeling is called a prime graph.
A graph with vertex set v is said to have a prime labeling if its vertices are labeled with distinct integers the labels assigned to x and y are relatively prime. Theory and applications graph labelings, where the vertices and edges are assigned, real values subject to certain conditions, have often been motivated by their utility to various applied fields and their intrinsic mathematical interest logico mathematical. For all other terminology and notations in graph theory i follow west 9. This book depicts graph labelings that have led to thoughtprovoking problems and conjectures. Some topics in graph theory the purpose of this book is to provide some results in a class of problems categorized as graph labeling. Pdf a graph g v, e with n vertices is said to admit prime labeling if its vertices can be labeled with distinct positive integers not exceeding n. Conferenceseminar papers in all areas of graph theory will be published as a special issue. In the following figures 3 and 4 prime labeling of a graph of order 5 and the prime labeling for the graph g1 obtained by identifying the vertices of with label 1 and 5 are shown. Prime labeling of families of trees with gaussian integers. They contain an introduction to basic concepts and results in graph theory, with a special emphasis put on the networktheoretic circuitcut dualism.
There are many kinds of graph labeling such as graceful labeling, magic labeling, prime labeling, and other different labeling techniques. The graph which admits prime labeling is called a prime graph. Buy graph theory book online at low prices in india graph. Graph theory issn and millions of other books are available for amazon kindle. Pdf prime labelling of some special graphs top journal. The most interesting problem is to prove the prime tree conjecture. This is a companion to the book introduction to graph theory world scientific, 2006. A graph on n vertices is said to admit a prime labeling if we can label its vertices with the first n natural numbers such that any two adjacent vertices have relatively prime labels. Acquaintanceship and friendship graphs describe whether people know each other. An introduction to the theory of knots computer graphics. The simplest example for the talk is the tetrahedron k 4. A graph which admits prime labeling is called a prime graph.
A graph consists of some points and lines between them. Graph labelings were rst introduced in the late 1960s. A primein a graph is a closed path in the graph minimizing the number of edges traversed. Graph labeling, prime labeling, square difference, prime graphs, planar graphs. Prime labeling for duplication of graph elements in k e. Rosa published a pioneering paper on graph labeling problems 36. The first graph in figure 4 is a path with six edges and it has an. Available at a lower price from other sellers that may not offer free prime shipping. Graph theory is also widely used in sociology as a way, for example, to measure actors prestige or to explore rumor spreading, notably through the use of social network analysis software. The primary aim of this book is to present a coherent introduction to graph theory, suitable as a textbook for advanced undergraduate and beginning graduate students in mathematics and computer science. In this paper, we consider when ladder graphs are prime and when the corresponding labeling may be done in a cyclic manner around. The vital application of labeled graphs can be found in science, engineering and technology and we refer 4 for. Thus in all the possibilities f1 admits a prime labeling for 1.
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