# graceful labelings

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GRACEFUL LABELINGS AND ITS APPLICATIONS

presentation by M.Radhika M.Phil MATHMATICS

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Over viewIntroductionDefinitionsVariations of Graceful Labeling Algorithms for Graceful LabelingApplicationsConclusion References

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introduction Graph labeling were first introduced in the late 1960s. Rosa [1967] defined a-valuations are functions that produce graceful labeling. However, the term graceful labeling was not used until Golomb studied such labeling several years later [1977].Acharya [1982] obtained that every graph can be embedded as an induced subgraph of a graceful graph. The graceful labeling problem is to determine which graphs are graceful. When studying graceful labeling, we consider only finite graphs. For all notations in graph theory we follow Harary [2001]. A. Solairaju and K. Chitra [2008] introduced a new concept of labeling called an Edge - Odd Graceful Labeling (EOGL).

definitions What is mean by Graph Labeling? If the vertices of the graph are assigned values subject to certain conditions than it is known as graph labeling. Most of the graph labeling problem will have following three common characteristics. A set of numbers from which vertex labels are chosen, A rule that assigns a value to each edge, A condition that these values must satisfy.

What is mean by Graceful Labeling?A graceful labeling is a labeling of the vertices of a graph with distinct integers from the set {0, 1, 2, ... ,q} (where q represents the number of edges) such that...if f(v) denotes the label even to vertex v, when each edge uv is given the value | f(u) f(v) |, the edges are labeled 1, 2, ... , q. 0 2 3

Are The Following Graphs Graceful? Path Graphs Cycle Graphs Complete Graphs Complete Bipartite Graphs Wheel Graphs Trees

Path Graphs

Theorem: Every path graph is graceful.

Theorem: Every path graph is graceful.

Path Graphs Proof: Let G be a path graph.Label the first vertex 0, and label every other vertex increasing by 1 each time.Label the second vertex q and label every other vertex decreasing by 1 each time.There are q + 1 vertices, so the first set will label its vertices with numbers from the set {0, 1, ... , q / 2} if q is even and from the set {0, 1, ... ,(q+1)/2} if q is odd. The second set will label its vertices with numbers from the set {(q+2)/2, ... , q} if q is even, and {(q+3)/2, ... , q} if q is odd. Thus, the vertices are labeled legally.

Path GraphsWith the vertices labeled in this manner, the edges attain the values q, q-1, q-2, ... 1, in that order.Thus, this is a graceful labeling, so G is graceful.Therefore, all path graphs are graceful.

VARIATIONS OF GRACEFUL LABELING

ALGORITHMS for graceful labelingExhaustive Labeling AlgorithmsForward-Thinking Labeling AlgorithmsApproximation Labeling Algorithms

applicationsx-ray crystallography: X-ray diffraction is one of the most powerful techniques for characterizing the structural properties of crystalline solids, in which a beam of X-rays strikes a crystal and diffracts into many specific directions. In some cases more than one structure has the same diffraction information. This problem is mathematically equivalent to determining all labeling of the appropriate graphs which produce a pre specified set of edge labels.

The communications networkaddressing A communication network is composed of nodes, each of which has computing power and can transmit and receive messages over communication links, wireless or cabled. The basic network topologies are include fully connected, mesh, star, ring, tree, bus. A single network may consist of several interconnected subnets of different topologies. These issues are discussed briefly in this paper. Networks are further classified as Local Area Networks (LAN), e.g. inside one building, or Wide Area Networks (WAN), e.g. between buildings. It might be useful to assign each user terminal a node label, subject to the constraint that all connecting edges (communication links) receive distinct labels. In this way, the numbers of any two communicating terminals automatically specify (by simple subtraction) the link label of the connecting path; and conversely, the path label uniquely specifies the pair of user terminals which it interconnects.

Automatic channel allocation for small wirelesslocal area networksThe interference can be avoided by means of a suitable channel assignment. The channel assignment problem is the problem to assign a channel nonnegative integer, to each TV or radio transmitters located at various places such that communication do not interfere.In interference graph the access points (vertices) are interfering with some other access points in the same region. The graph is called as interference graph, which is constructed by the access points as nodes. An undirected edge is connecting these nodes if the nodes interfere with each other when using the same channel. Now, the channel allocation problem is converted into graph labeling problem i.e. vertex labeling problem

Analyzing Communication Efficiency in sensornetworks with Voronoi GraphThe sensor networks have got variety of applications. Tracking of mobile objects, collection of environmental data, defense applications, health care etc, The sensor network is modeled as a graph to analyze the communication efficiency. Here voronoi graph is used to model the sensor network. Because voronoi graph is constructed in a plane in the form of polygons with nodes as the sensors and the Polygon boundaries can be considered as the sensing range of each sensor.

conclusion The main aim of this paper is to explore role of Graph Labeling in Communication field. Graph Labeling is powerful tool that makes things ease in various fields of networking as said above. An overview is presented especially to project the idea of Graph Labeling in graceful graph. Researches may get some information related to graceful labeling and its applications in communication field and can get some ideas related to their field of research.

Thank you

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