Embed Size (px)
Citation preview
GRACEFUL LABELINGS AND GRACEFUL LABELINGS AND ITS APPLICATIONS ITS APPLICATIONS
presentation by
MRadhika MPhil
MATHMATICS
Over viewOver view
Introduction Definitions Variations of Graceful Labeling Algorithms for Graceful
Labeling Applications Conclusion References
introductionintroduction Graph labeling Graph labeling were first introduced in the late 1960rsquos Rosa [1967] Rosa [1967] defined aβ-valuations are functions that produce graceful labeling However the term graceful labeling was not used until Golomb 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 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)
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
A rule that assigns a value to each edge
A condition that these values must satisfy
What is mean by Graceful LabelingA 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) ndash f(v) | the edges are labeled 1 2 q 0
2 3
Are The Following Are The Following Graphs GracefulGraphs Graceful
bull Path Graphsbull Cycle Graphsbull Complete Graphs
bull Complete Bipartite Graphs bull Wheel Graphs bull Trees
Path GraphsPath Graphs
Theorem Every path graph is graceful
Theorem Every path graph is graceful
Path Graphs Path Graphs Proof
Let G be a path graphLabel the first vertex 0 and label every other vertex increasing by 1 each timeLabel the second vertex q and label every other vertex decreasing by 1 each timeThere are q + 1 vertices so the first set will label itrsquos 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 itrsquos 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 GraphsPath Graphs
bull With the vertices labeled in this manner the edges attain the values q q-1 q-2 1 in that order
bull Thus this is a graceful labeling so G is gracefulbull Therefore all path graphs are graceful 1048576
VARIATIONS OF GRACEFUL VARIATIONS OF GRACEFUL LABELINGLABELING
ALGORITHMS for graceful labelingALGORITHMS for graceful labeling
Exhaustive Labeling AlgorithmsForward-Thinking Labeling AlgorithmsApproximation Labeling Algorithms
applicationsapplications
x-ray crystallographyx-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 networkThe communications networkaddressingaddressing
bull 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
bull These issues are discussed briefly in this paper Networks are further classified as Local Area Networks (LAN) eg inside one building or Wide Area Networks (WAN) eg between buildings It might be useful to assign each user terminal a ldquonode labelrdquo subject to the constraint that all connecting ldquoedgesrdquo (communication links) receive distinct labels
bull 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 Automatic channel allocation for small wirelesssmall 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 ndash 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 ie vertex labeling problem
Analyzing Communication Analyzing Communication Efficiency in sensorEfficiency in sensor
networks with Voronoi Graphnetworks with Voronoi GraphThe sensor networks have got variety of
applications Tracking of mobile objects collection of environmental data defense applications health care etchellip 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
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
Thank Thank youyou
Over viewOver view
Introduction Definitions Variations of Graceful Labeling Algorithms for Graceful
Labeling Applications Conclusion References
introductionintroduction Graph labeling Graph labeling were first introduced in the late 1960rsquos Rosa [1967] Rosa [1967] defined aβ-valuations are functions that produce graceful labeling However the term graceful labeling was not used until Golomb 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 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)
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
A rule that assigns a value to each edge
A condition that these values must satisfy
What is mean by Graceful LabelingA 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) ndash f(v) | the edges are labeled 1 2 q 0
2 3
Are The Following Are The Following Graphs GracefulGraphs Graceful
bull Path Graphsbull Cycle Graphsbull Complete Graphs
bull Complete Bipartite Graphs bull Wheel Graphs bull Trees
Path GraphsPath Graphs
Theorem Every path graph is graceful
Theorem Every path graph is graceful
Path Graphs Path Graphs Proof
Let G be a path graphLabel the first vertex 0 and label every other vertex increasing by 1 each timeLabel the second vertex q and label every other vertex decreasing by 1 each timeThere are q + 1 vertices so the first set will label itrsquos 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 itrsquos 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 GraphsPath Graphs
bull With the vertices labeled in this manner the edges attain the values q q-1 q-2 1 in that order
bull Thus this is a graceful labeling so G is gracefulbull Therefore all path graphs are graceful 1048576
VARIATIONS OF GRACEFUL VARIATIONS OF GRACEFUL LABELINGLABELING
ALGORITHMS for graceful labelingALGORITHMS for graceful labeling
Exhaustive Labeling AlgorithmsForward-Thinking Labeling AlgorithmsApproximation Labeling Algorithms
applicationsapplications
x-ray crystallographyx-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 networkThe communications networkaddressingaddressing
bull 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
bull These issues are discussed briefly in this paper Networks are further classified as Local Area Networks (LAN) eg inside one building or Wide Area Networks (WAN) eg between buildings It might be useful to assign each user terminal a ldquonode labelrdquo subject to the constraint that all connecting ldquoedgesrdquo (communication links) receive distinct labels
bull 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 Automatic channel allocation for small wirelesssmall 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 ndash 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 ie vertex labeling problem
Analyzing Communication Analyzing Communication Efficiency in sensorEfficiency in sensor
networks with Voronoi Graphnetworks with Voronoi GraphThe sensor networks have got variety of
applications Tracking of mobile objects collection of environmental data defense applications health care etchellip 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
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
Thank Thank youyou
introductionintroduction Graph labeling Graph labeling were first introduced in the late 1960rsquos Rosa [1967] Rosa [1967] defined aβ-valuations are functions that produce graceful labeling However the term graceful labeling was not used until Golomb 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 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)
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
A rule that assigns a value to each edge
A condition that these values must satisfy
What is mean by Graceful LabelingA 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) ndash f(v) | the edges are labeled 1 2 q 0
2 3
Are The Following Are The Following Graphs GracefulGraphs Graceful
bull Path Graphsbull Cycle Graphsbull Complete Graphs
bull Complete Bipartite Graphs bull Wheel Graphs bull Trees
Path GraphsPath Graphs
Theorem Every path graph is graceful
Theorem Every path graph is graceful
Path Graphs Path Graphs Proof
Let G be a path graphLabel the first vertex 0 and label every other vertex increasing by 1 each timeLabel the second vertex q and label every other vertex decreasing by 1 each timeThere are q + 1 vertices so the first set will label itrsquos 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 itrsquos 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 GraphsPath Graphs
bull With the vertices labeled in this manner the edges attain the values q q-1 q-2 1 in that order
bull Thus this is a graceful labeling so G is gracefulbull Therefore all path graphs are graceful 1048576
VARIATIONS OF GRACEFUL VARIATIONS OF GRACEFUL LABELINGLABELING
ALGORITHMS for graceful labelingALGORITHMS for graceful labeling
Exhaustive Labeling AlgorithmsForward-Thinking Labeling AlgorithmsApproximation Labeling Algorithms
applicationsapplications
x-ray crystallographyx-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 networkThe communications networkaddressingaddressing
bull 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
bull These issues are discussed briefly in this paper Networks are further classified as Local Area Networks (LAN) eg inside one building or Wide Area Networks (WAN) eg between buildings It might be useful to assign each user terminal a ldquonode labelrdquo subject to the constraint that all connecting ldquoedgesrdquo (communication links) receive distinct labels
bull 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 Automatic channel allocation for small wirelesssmall 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 ndash 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 ie vertex labeling problem
Analyzing Communication Analyzing Communication Efficiency in sensorEfficiency in sensor
networks with Voronoi Graphnetworks with Voronoi GraphThe sensor networks have got variety of
applications Tracking of mobile objects collection of environmental data defense applications health care etchellip 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
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
Thank Thank youyou
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
A rule that assigns a value to each edge
A condition that these values must satisfy
What is mean by Graceful LabelingA 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) ndash f(v) | the edges are labeled 1 2 q 0
2 3
Are The Following Are The Following Graphs GracefulGraphs Graceful
bull Path Graphsbull Cycle Graphsbull Complete Graphs
bull Complete Bipartite Graphs bull Wheel Graphs bull Trees
Path GraphsPath Graphs
Theorem Every path graph is graceful
Theorem Every path graph is graceful
Path Graphs Path Graphs Proof
Let G be a path graphLabel the first vertex 0 and label every other vertex increasing by 1 each timeLabel the second vertex q and label every other vertex decreasing by 1 each timeThere are q + 1 vertices so the first set will label itrsquos 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 itrsquos 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 GraphsPath Graphs
bull With the vertices labeled in this manner the edges attain the values q q-1 q-2 1 in that order
bull Thus this is a graceful labeling so G is gracefulbull Therefore all path graphs are graceful 1048576
VARIATIONS OF GRACEFUL VARIATIONS OF GRACEFUL LABELINGLABELING
ALGORITHMS for graceful labelingALGORITHMS for graceful labeling
Exhaustive Labeling AlgorithmsForward-Thinking Labeling AlgorithmsApproximation Labeling Algorithms
applicationsapplications
x-ray crystallographyx-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 networkThe communications networkaddressingaddressing
bull 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
bull These issues are discussed briefly in this paper Networks are further classified as Local Area Networks (LAN) eg inside one building or Wide Area Networks (WAN) eg between buildings It might be useful to assign each user terminal a ldquonode labelrdquo subject to the constraint that all connecting ldquoedgesrdquo (communication links) receive distinct labels
bull 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 Automatic channel allocation for small wirelesssmall 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 ndash 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 ie vertex labeling problem
Analyzing Communication Analyzing Communication Efficiency in sensorEfficiency in sensor
networks with Voronoi Graphnetworks with Voronoi GraphThe sensor networks have got variety of
applications Tracking of mobile objects collection of environmental data defense applications health care etchellip 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
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
Thank Thank youyou
What is mean by Graceful LabelingA 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) ndash f(v) | the edges are labeled 1 2 q 0
2 3
Are The Following Are The Following Graphs GracefulGraphs Graceful
bull Path Graphsbull Cycle Graphsbull Complete Graphs
bull Complete Bipartite Graphs bull Wheel Graphs bull Trees
Path GraphsPath Graphs
Theorem Every path graph is graceful
Theorem Every path graph is graceful
Path Graphs Path Graphs Proof
Let G be a path graphLabel the first vertex 0 and label every other vertex increasing by 1 each timeLabel the second vertex q and label every other vertex decreasing by 1 each timeThere are q + 1 vertices so the first set will label itrsquos 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 itrsquos 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 GraphsPath Graphs
bull With the vertices labeled in this manner the edges attain the values q q-1 q-2 1 in that order
bull Thus this is a graceful labeling so G is gracefulbull Therefore all path graphs are graceful 1048576
VARIATIONS OF GRACEFUL VARIATIONS OF GRACEFUL LABELINGLABELING
ALGORITHMS for graceful labelingALGORITHMS for graceful labeling
Exhaustive Labeling AlgorithmsForward-Thinking Labeling AlgorithmsApproximation Labeling Algorithms
applicationsapplications
x-ray crystallographyx-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 networkThe communications networkaddressingaddressing
bull 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
bull These issues are discussed briefly in this paper Networks are further classified as Local Area Networks (LAN) eg inside one building or Wide Area Networks (WAN) eg between buildings It might be useful to assign each user terminal a ldquonode labelrdquo subject to the constraint that all connecting ldquoedgesrdquo (communication links) receive distinct labels
bull 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 Automatic channel allocation for small wirelesssmall 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 ndash 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 ie vertex labeling problem
Analyzing Communication Analyzing Communication Efficiency in sensorEfficiency in sensor
networks with Voronoi Graphnetworks with Voronoi GraphThe sensor networks have got variety of
applications Tracking of mobile objects collection of environmental data defense applications health care etchellip 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
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
Thank Thank youyou
Are The Following Are The Following Graphs GracefulGraphs Graceful
bull Path Graphsbull Cycle Graphsbull Complete Graphs
bull Complete Bipartite Graphs bull Wheel Graphs bull Trees
Path GraphsPath Graphs
Theorem Every path graph is graceful
Theorem Every path graph is graceful
Path Graphs Path Graphs Proof
Let G be a path graphLabel the first vertex 0 and label every other vertex increasing by 1 each timeLabel the second vertex q and label every other vertex decreasing by 1 each timeThere are q + 1 vertices so the first set will label itrsquos 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 itrsquos 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 GraphsPath Graphs
bull With the vertices labeled in this manner the edges attain the values q q-1 q-2 1 in that order
bull Thus this is a graceful labeling so G is gracefulbull Therefore all path graphs are graceful 1048576
VARIATIONS OF GRACEFUL VARIATIONS OF GRACEFUL LABELINGLABELING
ALGORITHMS for graceful labelingALGORITHMS for graceful labeling
Exhaustive Labeling AlgorithmsForward-Thinking Labeling AlgorithmsApproximation Labeling Algorithms
applicationsapplications
x-ray crystallographyx-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 networkThe communications networkaddressingaddressing
bull 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
bull These issues are discussed briefly in this paper Networks are further classified as Local Area Networks (LAN) eg inside one building or Wide Area Networks (WAN) eg between buildings It might be useful to assign each user terminal a ldquonode labelrdquo subject to the constraint that all connecting ldquoedgesrdquo (communication links) receive distinct labels
bull 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 Automatic channel allocation for small wirelesssmall 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 ndash 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 ie vertex labeling problem
Analyzing Communication Analyzing Communication Efficiency in sensorEfficiency in sensor
networks with Voronoi Graphnetworks with Voronoi GraphThe sensor networks have got variety of
applications Tracking of mobile objects collection of environmental data defense applications health care etchellip 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
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
Thank Thank youyou
Path GraphsPath Graphs
Theorem Every path graph is graceful
Theorem Every path graph is graceful
Path Graphs Path Graphs Proof
Let G be a path graphLabel the first vertex 0 and label every other vertex increasing by 1 each timeLabel the second vertex q and label every other vertex decreasing by 1 each timeThere are q + 1 vertices so the first set will label itrsquos 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 itrsquos 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 GraphsPath Graphs
bull With the vertices labeled in this manner the edges attain the values q q-1 q-2 1 in that order
bull Thus this is a graceful labeling so G is gracefulbull Therefore all path graphs are graceful 1048576
VARIATIONS OF GRACEFUL VARIATIONS OF GRACEFUL LABELINGLABELING
ALGORITHMS for graceful labelingALGORITHMS for graceful labeling
Exhaustive Labeling AlgorithmsForward-Thinking Labeling AlgorithmsApproximation Labeling Algorithms
applicationsapplications
x-ray crystallographyx-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 networkThe communications networkaddressingaddressing
bull 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
bull These issues are discussed briefly in this paper Networks are further classified as Local Area Networks (LAN) eg inside one building or Wide Area Networks (WAN) eg between buildings It might be useful to assign each user terminal a ldquonode labelrdquo subject to the constraint that all connecting ldquoedgesrdquo (communication links) receive distinct labels
bull 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 Automatic channel allocation for small wirelesssmall 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 ndash 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 ie vertex labeling problem
Analyzing Communication Analyzing Communication Efficiency in sensorEfficiency in sensor
networks with Voronoi Graphnetworks with Voronoi GraphThe sensor networks have got variety of
applications Tracking of mobile objects collection of environmental data defense applications health care etchellip 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
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
Thank Thank youyou
Path Graphs Path Graphs Proof
Let G be a path graphLabel the first vertex 0 and label every other vertex increasing by 1 each timeLabel the second vertex q and label every other vertex decreasing by 1 each timeThere are q + 1 vertices so the first set will label itrsquos 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 itrsquos 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 GraphsPath Graphs
bull With the vertices labeled in this manner the edges attain the values q q-1 q-2 1 in that order
bull Thus this is a graceful labeling so G is gracefulbull Therefore all path graphs are graceful 1048576
VARIATIONS OF GRACEFUL VARIATIONS OF GRACEFUL LABELINGLABELING
ALGORITHMS for graceful labelingALGORITHMS for graceful labeling
Exhaustive Labeling AlgorithmsForward-Thinking Labeling AlgorithmsApproximation Labeling Algorithms
applicationsapplications
x-ray crystallographyx-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 networkThe communications networkaddressingaddressing
bull 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
bull These issues are discussed briefly in this paper Networks are further classified as Local Area Networks (LAN) eg inside one building or Wide Area Networks (WAN) eg between buildings It might be useful to assign each user terminal a ldquonode labelrdquo subject to the constraint that all connecting ldquoedgesrdquo (communication links) receive distinct labels
bull 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 Automatic channel allocation for small wirelesssmall 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 ndash 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 ie vertex labeling problem
Analyzing Communication Analyzing Communication Efficiency in sensorEfficiency in sensor
networks with Voronoi Graphnetworks with Voronoi GraphThe sensor networks have got variety of
applications Tracking of mobile objects collection of environmental data defense applications health care etchellip 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
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
Thank Thank youyou
Path GraphsPath Graphs
bull With the vertices labeled in this manner the edges attain the values q q-1 q-2 1 in that order
bull Thus this is a graceful labeling so G is gracefulbull Therefore all path graphs are graceful 1048576
VARIATIONS OF GRACEFUL VARIATIONS OF GRACEFUL LABELINGLABELING
ALGORITHMS for graceful labelingALGORITHMS for graceful labeling
Exhaustive Labeling AlgorithmsForward-Thinking Labeling AlgorithmsApproximation Labeling Algorithms
applicationsapplications
x-ray crystallographyx-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 networkThe communications networkaddressingaddressing
bull 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
bull These issues are discussed briefly in this paper Networks are further classified as Local Area Networks (LAN) eg inside one building or Wide Area Networks (WAN) eg between buildings It might be useful to assign each user terminal a ldquonode labelrdquo subject to the constraint that all connecting ldquoedgesrdquo (communication links) receive distinct labels
bull 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 Automatic channel allocation for small wirelesssmall 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 ndash 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 ie vertex labeling problem
Analyzing Communication Analyzing Communication Efficiency in sensorEfficiency in sensor
networks with Voronoi Graphnetworks with Voronoi GraphThe sensor networks have got variety of
applications Tracking of mobile objects collection of environmental data defense applications health care etchellip 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
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
Thank Thank youyou
VARIATIONS OF GRACEFUL VARIATIONS OF GRACEFUL LABELINGLABELING
ALGORITHMS for graceful labelingALGORITHMS for graceful labeling
Exhaustive Labeling AlgorithmsForward-Thinking Labeling AlgorithmsApproximation Labeling Algorithms
applicationsapplications
x-ray crystallographyx-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 networkThe communications networkaddressingaddressing
bull 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
bull These issues are discussed briefly in this paper Networks are further classified as Local Area Networks (LAN) eg inside one building or Wide Area Networks (WAN) eg between buildings It might be useful to assign each user terminal a ldquonode labelrdquo subject to the constraint that all connecting ldquoedgesrdquo (communication links) receive distinct labels
bull 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 Automatic channel allocation for small wirelesssmall 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 ndash 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 ie vertex labeling problem
Analyzing Communication Analyzing Communication Efficiency in sensorEfficiency in sensor
networks with Voronoi Graphnetworks with Voronoi GraphThe sensor networks have got variety of
applications Tracking of mobile objects collection of environmental data defense applications health care etchellip 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
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
Thank Thank youyou
ALGORITHMS for graceful labelingALGORITHMS for graceful labeling
Exhaustive Labeling AlgorithmsForward-Thinking Labeling AlgorithmsApproximation Labeling Algorithms
applicationsapplications
x-ray crystallographyx-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 networkThe communications networkaddressingaddressing
bull 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
bull These issues are discussed briefly in this paper Networks are further classified as Local Area Networks (LAN) eg inside one building or Wide Area Networks (WAN) eg between buildings It might be useful to assign each user terminal a ldquonode labelrdquo subject to the constraint that all connecting ldquoedgesrdquo (communication links) receive distinct labels
bull 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 Automatic channel allocation for small wirelesssmall 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 ndash 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 ie vertex labeling problem
Analyzing Communication Analyzing Communication Efficiency in sensorEfficiency in sensor
networks with Voronoi Graphnetworks with Voronoi GraphThe sensor networks have got variety of
applications Tracking of mobile objects collection of environmental data defense applications health care etchellip 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
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
Thank Thank youyou
applicationsapplications
x-ray crystallographyx-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 networkThe communications networkaddressingaddressing
bull 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
bull These issues are discussed briefly in this paper Networks are further classified as Local Area Networks (LAN) eg inside one building or Wide Area Networks (WAN) eg between buildings It might be useful to assign each user terminal a ldquonode labelrdquo subject to the constraint that all connecting ldquoedgesrdquo (communication links) receive distinct labels
bull 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 Automatic channel allocation for small wirelesssmall 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 ndash 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 ie vertex labeling problem
Analyzing Communication Analyzing Communication Efficiency in sensorEfficiency in sensor
networks with Voronoi Graphnetworks with Voronoi GraphThe sensor networks have got variety of
applications Tracking of mobile objects collection of environmental data defense applications health care etchellip 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
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
Thank Thank youyou
The communications networkThe communications networkaddressingaddressing
bull 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
bull These issues are discussed briefly in this paper Networks are further classified as Local Area Networks (LAN) eg inside one building or Wide Area Networks (WAN) eg between buildings It might be useful to assign each user terminal a ldquonode labelrdquo subject to the constraint that all connecting ldquoedgesrdquo (communication links) receive distinct labels
bull 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 Automatic channel allocation for small wirelesssmall 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 ndash 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 ie vertex labeling problem
Analyzing Communication Analyzing Communication Efficiency in sensorEfficiency in sensor
networks with Voronoi Graphnetworks with Voronoi GraphThe sensor networks have got variety of
applications Tracking of mobile objects collection of environmental data defense applications health care etchellip 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
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
Thank Thank youyou
Automatic channel allocation for Automatic channel allocation for small wirelesssmall 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 ndash 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 ie vertex labeling problem
Analyzing Communication Analyzing Communication Efficiency in sensorEfficiency in sensor
networks with Voronoi Graphnetworks with Voronoi GraphThe sensor networks have got variety of
applications Tracking of mobile objects collection of environmental data defense applications health care etchellip 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
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
Thank Thank youyou
Analyzing Communication Analyzing Communication Efficiency in sensorEfficiency in sensor
networks with Voronoi Graphnetworks with Voronoi GraphThe sensor networks have got variety of
applications Tracking of mobile objects collection of environmental data defense applications health care etchellip 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
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
Thank Thank youyou
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
Thank Thank youyou
Thank Thank youyou
