This document discusses graceful graph labelings and their applications. It begins with an introduction that defines graph labelings and graceful labelings. It then examines whether specific graph types like paths, cycles, and trees admit graceful labelings. The document outlines algorithms for determining graceful labelings and describes applications in fields like communication networks, sensor networks, and X-ray crystallography where determining graceful labelings can help address problems like channel allocation and network addressing. It concludes by emphasizing how graph labelings are useful tools in communication-related domains.

1. GRACEFUL LABELINGSGRACEFUL LABELINGS
AND ITS APPLICATIONSAND ITS APPLICATIONS
presentation
by
M.Radhika

2. Over viewOver view
 Introduction
 Definitions
 Variations of Graceful Labeling
 Algorithms for Graceful Labeling
 Applications
 Conclusion
 References

3. intrOductiOnintrOductiOn
Graph labelingGraph labeling were first introduced in the late 1960’s.
Rosa [1967]Rosa [1967] defined aβ-valuations are functions that
produce graceful labeling. However, the term graceful
labeling was not used until GolombGolomb 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 graphsfinite 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).

4. definitiOnsdefinitiOns
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,

5. 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

6. Are The FollowingAre The Following
Graphs Graceful?Graphs Graceful?
• Path Graphs
• Cycle Graphs
• Complete Graphs
• Complete Bipartite Graphs
• Wheel Graphs
• Trees

7. Path GraphsPath Graphs
Theorem: Every path graph is graceful.

8. Path GraphsPath 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
it’s 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 it’s 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.

9. Path GraphsPath Graphs
• With 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. �

10. VARIATIONS OF GRACEFULVARIATIONS OF GRACEFUL
LABELINGLABELING

11. ALGORITHMS fOR GRAcefuLALGORITHMS fOR GRAcefuL
LAbeLInGLAbeLInG
 Exhaustive Labeling Algorithms
 Forward-Thinking Labeling Algorithms
 Approximation Labeling Algorithms

12. AppLIcATIOnSAppLIcATIOnS
x-ray crystallography:x-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.

13. The communications networkThe communications network
addressingaddressing
• 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

14. Automatic channel allocation for small wirelessAutomatic channel allocation for small wireless
local area networkslocal area networks
 The 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

15. Analyzing Communication Efficiency in sensorAnalyzing Communication Efficiency in sensor
networks with Voronoi Graphnetworks with Voronoi Graph
The 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.

16. conclusionconclusion
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.

17. Thank
Thank
youyou