project on graph theory

7/29/2019 Project on Graph Theory

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A PROJECT ON GRAPH THEORY AND ITS APPLICATIONS

PARTIAL FULFILLMENT OF THE REQUIREMENT FOR THE

DEGREE OF

MATHEMATICS OF SCIENCE (PART II)

IN MATHEMATICS

CAM PRACTICAL, PAPER (XIV)

PRESENTED BY

NAME ROLL NO.

Sheet Nihal Topno 09MS5496591

Binesh Kumar Yadav 09MS5496597

Ratan Kumar Mishra 09MS5496620

Tahsin Ahmad09MS5496591

Anup Kumar Keshri 09MS5496589

Surya Narayan Pradhan 09MS549660

Md. Aslam Ansari 09MS5496626

Shamsad Ahmad 09MS5496591

(Group A)


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CERTIFICATION

The project titled The Project Report on Graph Theory and Its

Applications is the work done by Group A is hereby approved

as a creditable work and presented in a manner satisfactory to

warrant its acceptance as its serve the purpose for which it has

been submitted.

SIGNATURE

(GUIDE)

INTERNAL EXAMINER EXTERNAL EXAMINER

HEAD OF THE DEPARTMENT

DEPARTMENT OF MATHEMATICS

RANCHI UNIVERSITY, RANCHI


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ACNOWLEDGEMENT

The successful completion of the project is solely due to

the guidance and encouragement received from the peoplearound us. We would sincerely like to thank to all of them.

We have received constant support from the cooperative

and efficient faculty members of our department and so we

would like to thank them.

Our special thank to our H. O. D. Dr. N. K. Aggrawal, our

project guide Mr. Aamir Khusro for their help and support.

Finally, we would like to thank to entire Department of

Mathematics, Ranchi University, Ranchi for their help in course

of development of every phase of our project.

Sheet Nihal Topno

Binesh Kumar Yadav

Ratan Kumar Mishra

Tahsin Ahmed

Anup Keshri

Md. Aslam Ansari

SuryaNarayan Pradhan

Shamshad Ahmad


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GRAPH THEORY

INTRODUCTION

There is an interesting story behind the development of

Graph Theory. A river Pregel (Preger) flows through the city of

Knigsberg, thereby forming two islands in the city. These

islands were connected by seven bridges as shown in figure

Fig. 1

The citizens of Knigsberg had an entertaining

problem. Is it possible to cross each one of seven bridges exactly

once and come back to starting point? This problem is known as

Knigsberg Bridge Problem.

Euler was fascinated by the problem and it was he who

gave the mathematical proof for the impossibility of such a rule.

Euler represented each land area by a point and joined the two


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such points by a line which was symbol for the bridge. His

representation was

Fig. 2

This is nothing but a mapping a real world problem into

a paper. Various networks like road map, electric network air

routes are represented in this way. Euler worked on the problem

and developed new branch of mathematics called graph theory.

With the help of the algorithms and theorems of graph theory

many problems of real world are solved. Many mathematicians

have also worked on this topic and given mathematical theorems

and algorithms.

Euler was one who translated a real world problem into

a mathematical problem. In this project we present the way this

mathematical problem can be transferred to a computer. We

also provide a few C program code to solve some problems of

graph theory with the help of computer easily.

Mathematically such a representation is called a graph.

Mathematical definitions and properties of graph are discussed

further.


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Definition:-A graph G is a set of two tuples (V,E), where V

represents the non empty set of vertices of G and E is a subset of

VV called edge set of G.

Fig. 3

Directed Graph: A graph G=(V,E) is said to be a directed graph

if its every edge ek=(vi,vj) is represented as a line segment from

vertex vi to vjwith an arrow directed from vi to vj.

Fig. 4


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Weighted Graph: Let G be a graph. For any edge e, w(e) called

the weight of e, is a real number. If each edge of the graph is

assigned with some real number then G is called a weighted

graph.

Fig. 5

Fig. 7


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REPRESENTATION OF GRAPH IN COMPUTER

Usually the first step in representing a graph is to map

the vertices to a set of contiguous integers. (0,|V|-1) is the most

convenient in C- programs. There are two popular ways that are

used to maintain a graph in a computer memory. These are:

1. Sequential

2. Linked list

Sequential Representation: The graphs can be represented as

matrices in sequential representation. There are two most

common matrices. These are:

1. Adjacency

2. Incidence

The adjacency matrix is a sequence matrix with one row

and one column for each vertex. The entries of the matrix are

either 0 or 1. A value of 1 for row i and column j implies that edge

eij exists between vi and vj vertices. A value of 0 implies that

there is no edge between vi and vj. In other words we can say

that if graph G consists of v1, v2, v3,..,vn vertices then the

adjacency matrix A=[aij] of graph G is the nn matrix and can be

defined as:

1 If vi is adjacent to vj (that is if there is an

aij = edge between vi and vj)

0 If there is no edge between vi and vj.


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Fig.7

Such a matrix A. which contains entries of only 0 or 1, is

called a bit matrix or a Boolean. For example consider the graph

G illustrated in above figure, which consists of 5 vertices and 10

edges. The set of vertices is as follows:

V= { v1, v2, v3, v4, v5}and the set of edges is:

E= {(v1, v2), (v1, v3), (v2, v1), (v2, v3), (v2, v5), (v3, v1), (v3, v4), (v4, v1),

(v4, v5), (v5, v3)}

The adjacency matrix A for G is as follows:

v1 v2 v3 v4 v5

v1 0 1 1 0 0

v2 1 0 1 0 1

A= v3 1 0 0 1 0

v4 1 0 0 0 1

v5 0 0 1 0 0


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From the matrix A, it is clear that number of 1s in the

adjacency matrix is equal to the number of edges in the graph.

When one is constructing an adjacency matrix for a graph, one

must follow the following points:

1.Adjacency matrix does not depend on the ordering ofvertex of a graph G, i.e. different ordering of vertices may

result in a different adjacency matrix. One can obtain the

same matrix by interchanging rows and columns.

2. If the graph G is undirected then the adjacency matrix of Gwill be symmetric. i.e. [aij]=[aji] for every i and j.

The incidence matrix consists of a row for every vertex

and a column for every edge. The entries of the matrix are -1, 0

and 1. If kth edge is (vi ,vj), the kth column has a value 1 in ith row, -

1 in the jth row and 0 elsewhere. For example, the incidence

matrix I for the above graph is as follows:

1 1 -1 0 0 -1 0 -1 0 0

-1 0 1 1 1 0 0 0 0 0

I= 0 -1 0 -1 0 1 1 0 0 -1

0 0 0 0 0 0 -1 1 1 0

0 0 0 0 -1 0 0 0 -1 1


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/* Program for creation of adjacency matrix of a

simple graph */

#include

#include

#define max 20

int adj[max][max]; /*Adjacency matrix */

int n; /* Denotes number of nodes in the graph */

main()

{

int max_edges,i,j,origin,destin;

char graph_type;

printf("Enter number of vertices : ");

scanf("%d",&n);

printf("Enter type of graph,directed/undirected (d/u):");

fflush(stdin);

scanf("%c",&graph_type);

if(graph_type=='u')

max_edges=n*(n-1);

else

max_edges=n*(n-1)/2;


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for(i=1;i n || destin> n || origin


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printf("The adjacency matrix is :\n");

for(i=1;i


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OUTPUT

Adjacency matrix of the following undirected graphthrough c programming :

Enter number of edges : 4

Enter type of graph, directed or undirected (d/u): u

Enter edge 1 (end vertices):

1

2

Enter edge 2 (end vertices):1

3


1

4


1


2

3



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3

1


3

2


3

4


4

1


4

3


2


1

2

The adjacency matrix is :0 1 1 1

1 0 1 0

1 1 0 1

1 0 1 0


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OUTPUT

Adjacency matrix of the following directed graphthrough c programming :

Enter number of edges : 4

Enter type of graph, directed or undirected (d/u): d


1

2


2

3


3

1


1

4


4


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3


12

The adjacency matrix is :

0 1 -1 1

-1 0 1 0

1 -1 0 -1-1 0 1 0


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DEFINITIONS AND THEOREMS OF GRAPH THEORY

Loop: An edge of the form (a, a), a belonging to V, vertex set ofG is called a loop.

Fig.8

Multiple Edges: Two edges e1 and e2 of a graph G are said to be

multiple edges if they have same end vertices.

.

Fig.9


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Null Graph: A null graph is a graph which has no edges.

Fig. 10

A vertex in a graph G which is not connected to the other

vertices of the graph by an edge is called an isolated vertex. All

the vertices of null graph are isolated.

Adjacency matrix for a null graph is of the form:

v1 v2 v3 . vn

v1 0 0 0 . 0

v2 0 0 0 . 0

v3 0 0 0 . 0

... 0 0 0 . 0

vn o 0 0 . 0


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Complete Graph: A simple graph is said to be complete if each

pair of vertices of the graph are adjacent to each other.

Fig. 11

Adjacency matrix for a graph having no loop is of the form:

v1 v2 v3 v4 ........... vn

v1 0 1 1 1 .. 1

v2 1 0 1 1 1

v3 1 1 0 1 1

...

vn 1 1 1 1 ... 0

Simple Graph: A Graph G is said to be simple if it has no loop as

well as multiple edges.

Simple graphs play an important role in further

discussion of graph theory.


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Cycle Graph: The cycle graph is a graph with vertices v1, v2, ..,

vn whose edge set is (v1 ,v2), (v2,v3),., (vn-1, vn), (vn, v1) with a

particular order of vertices.

Fig. 12

Bipartite Graph: A graph G(V,E) is said to be bipartite if the

vertex set can be split into two non-empty subsets X and Y such

that X Y= and XY=V and for any edge (a, b)E either a X

& b Y or a Y & b X.

Fig. 13


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In the graph of Fig.12 the vertices can be partitioned into X{v1,

v6, v

8, v

3} and Y{v

2, v

4, v

5,v

7}. Every edge in the graph has one end

in X and another end in Y which can be easily inspected.

A graph G is bipartite if and only if each cycle of G ifexists is of even length.

Complete bipartite graph: A bipartite graph G with partitions

X and Y is said to be complete if each vertex of X is adjacent to

each vertex of Y.

Fig. 14

In the graph of Fig. 14 the set of vertices can be

partitioned into X{v1, v2, v3 } and Y{v3, v4, v5} so that each vertex ifthe graph has an end in both the sets. Also each vertex of X is

connected with each vertex of Y, making the graph bipartite.

Degree of vertex: Let V be any vertex of a graph G. The degree

of v is denoted by deg(v) or d(v) and defined as deg(v)=number

of edge end associated with v in G.


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Even or Odd degree vertex: A vertex v of a graph G is said to

be even or odd according as the degree of the vertex is even or

odd.

The number of odd vertices in a graph G is even. This fact isused in handshaking lemma.

Subgraph: A graph H(V(H),E(H)) is said to be a subgraph of

G(V(G),E(G)) if V(H)V(G) and E(H)E(G).

Fig. 15(a) Fig. 15(b)

Component of a graph: A component K of a graph is a

subgraph of G in which there exists a path between each pair of

vertices of K.

Connected Graph: A graph G is said to be connected if thereexists a path from a vertex to another vertex to another vertex of

the graph.

If G is a simple graph n vertices, m edges and k componentsthen n-k m (n-k)(n-k+1)/2.

A simple graph with n vertices and more than (n-1)(n-2)/2edges is connected.


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Cut Edge(Bridge): A cut edge of a graph G is an edge of G after

removal of it the graph G becomes disconnected.

Fig.16

The graph given in Fig.16 edges (v3,v4) and (v6, v7) are

cut edges (bridges).

An edge e of a graph G is a cut edge of G if and only of e iscontained in no cycle of G.

Cut Set: A cut set of a graph G is a subset of edge set of G such

that after removal of all the edges of the set the graph G becomes

disconnected.


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PATHS AND CYCLES

Walk:Any sequence of edges of a graph such that the terminal

vertex of an edge is the initial vertex of the edge appearing next

to it is called a walk.

Path: A walk is said to be a path if there is no vertices with

repetation except possibly end vertex.

Trail:A walk is said to be trail if there are no edges with

repetation.

Connected graph: A graph G is said to be connected if there

exists a path from a vertex to another vertex of the graph.

Cycle: A closed path in a graph G is said to be a cycle.

If G is a graph with minimum degree of vertices of G, (G)2, then G contains a cycle of length at least +1.

If G is a graph with p edges and q vertices such that qp,then G contains a cycle.

Eulerian Trail:A closed trail containing all the edges of a graph

G is called anEulerian trail of the graph.

Eulerian Graph: A graph G is said to be Eulerian if it contains

an Eulerian trail.

Euler finally proposed a theorem to show the

impossibility of occurance of the rule discussed previously.

Furthermore his theorem goes on to say that such rule is

impossible for every graph having a certain property. If such a

rule is possible for a graph then we call that graph Eulerian.


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Eulers theorem: A connected graph G is said to be Eulerian if

and only if degree of each vertex of G is even.

Euler considered a problem which assumes no

repetation of edges however vertices can be repeated. We now

consider a modified case in which vertices are non repetative.

Hamiltonian cycle: A cycle containing all the vertices of a

graph G is called a Hamiltonian cycle of G.

Hamiltonian Graph: A graph is said to be Hamiltonian if it

contains a Hamiltonian cycle.

Following theorem of Dirac is very important in order

to check whether a given graph is hamiltonian or not.

Let G be a simple connected graph with n (3) vertices and(u)n/2 for each vertex u in G, then G is Hamiltoian.


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TREE

Cyclic Graph: A graph is said to be cyclic if it contains a cycle.

Tree: A connected acyclic graph is called a tree.

Fig. 17In any tree two vertices are connected by a unique

path.

In a tree number of edges is one less than number ofvertices.

Every non-trivial tree has at least two vertices ofdegree one.

A connected graph is a tree if and only if every edge isa cut set.

If G is a graph with vertices and v-1 edges thenfollowing statements are equivalent: (a) G is

connected, (b) G is acyclic, (c) G is a tree.

Forest: A graph having no cycle is called forest.


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Spanning tree: A spanning tree of a connected graph G is a

subgraph of G which is a tree containing all the vertices of G.

Every connected graph contains a spanning tree.Minimum Spanning Tree: A graph G may have many spanning

tries. If weigth is associated with each edge of G then the

spanning tree of minimum weigth is called the minimun

spanning tree of graph G.

There are a few algorithms to find out minimum spanning

tree of a graph. We disscuss Kruskals Algorithm and present Cprogramming code for this algorithm.

Kruskals Algorithm:

Step 1: Choose e1 an edge of G such that w(e1) is small as possible

and e1 is not a loop.

Step 2: If edges e1, e2,.,ei have been chosen then choose an edge

ei+1 not already chosen such that (i) the induced graph

G{e1, e2,.., ei, ei+1} is acyclic. (ii) weight of ei+1 is as small as

possible.

Step 3: If G has n vertices stop after n-1 edges have chosen

otherwise repeat step 2.

Step 4: Exit.


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/*Program for creating a minimum spanning tree

from Kruskal's algorithm*/

#include

#include

#include

#define MAX 20

struct edge

{

int u;

int v;

int weight;

struct edge *link;

}*front = NULL;

int father[MAX]; /*Holds father of each node */

struct edge tree[MAX]; /* Will contain the edges of spanning

tree */

int n; /*Denotes total number of nodes in the graph */

int wt_tree=0; /*Weight of the spanning tree */

int count=0; /* Denotes number of edges included in the tree */


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/* Functions */

void make_tree();

void insert_tree(int i,int j,int wt);

void insert_pque(int i,int j,int wt);

struct edge *del_pque();

create_graph();

main()

{

int i;

create_graph();

make_tree();

printf("Edges to be included in spanning tree are :\n");

for(i=1;i


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create_graph()

{

int i,wt,max_edges,origin,destin;


scanf("%d",&n);

max_edges=n*(n-1)/2;

for(i=1;i n || destin > n || origin


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}/*End of for*/

if(i==0)

{

printf("Spanning tree is not possible\n");

}

}/*End of create_graph()*/

void make_tree()

{

struct edge *tmp;

int node1,node2,root_n1,root_n2;

while( count < n-1) /*Loop till n-1 edges included in thetree*/

{

tmp=del_pque();

node1=tmp->u;

node2=tmp->v;

printf("n1=%d ",node1);

printf("n2=%d ",node2);

while( node1 > 0)


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{

root_n1=node1;

node1=father[node1];

}

while( node2 >0 )

{

root_n2=node2;

node2=father[node2];

}

printf("rootn1=%d ",root_n1);

printf("rootn2=%d\n",root_n2);

if(root_n1!=root_n2)

{

insert_tree(tmp->u,tmp->v,tmp->weight);

wt_tree=wt_tree+tmp->weight;

father[root_n2]=root_n1;

}

}/*End of while*/

}/*End of make_tree()*/

/*Inserting an edge in the tree */


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void insert_tree(int i,int j,int wt)

{

printf("This edge is inserted in the spanning tree\n");

count++;

tree[count].u=i;

tree[count].v=j;

tree[count].weight=wt;

}/*End of insert_tree()*/

/*Inserting edges in the priority queue */

void insert_pque(int i,int j,int wt)

{

struct edge *tmp,*q;

tmp = (struct edge *)malloc(sizeof(struct edge));

tmp->u=i;

tmp->v=j;

tmp->weight = wt;

/*Queue is empty or edge to be added has weight less than first

edge*/

if( front == NULL || tmp->weight < front->weight )


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{

tmp->link = front;

front = tmp;

}

else

{

q = front;

while( q->link != NULL && q->link->weight weight )

q=q->link;

tmp->link = q->link;

q->link = tmp;if(q->link == NULL) /*Edge to be added at the end*/

tmp->link = NULL;

}/*End of else*/

}/*End of insert_pque()*/

/*Deleting an edge from the priority queue*/

struct edge *del_pque()

{

struct edge *tmp;


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tmp = front;

printf("Edge processed is %d->%d %d\n",tmp->u,tmp-

>v,tmp->weight);

front = front->link;

return tmp;

}/*End of del_pque()*/


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OUTPUT

The minimum spanning tree of the following graphthrough c programming :

Enter number of vertices : 5

Enter edge 1 (end vertices) : 1

2

Enter weight for this edge : 2


3



4



5



1


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2Enter edge 6 (end vertices) : 3


5


5Enter weight for this edge : 2


4



3



2


Edge processed is 4->5 1

n1=4 n2=5 rootn1=4 rootn2=5

This edge is inserted in the spanning tree




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This edge is inserted in the spanning treeEdge processed is 3->4 2


This edge is inserted in the spanning tree


n1=5 n2=1 rootn1=3 rootn2=1This edge is inserted in the spanning tree

Edges to be inserted in the spanning tree are :

4->5

1->2

3->4

5->1

Weight of this minimum spanning tree is : 7

Hence the minimum spanning tree of above graph is :


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PLANARITY

Planar Graph:A graph G is said to be plannar if it can be drawn

in a plane without intersecting the edges except its end vertices.

Every simple planar graph contains a vertex of degreeatmost five.

Graph K5 (Complete graph in 5 vertices) and K3,3(a completebipartite graph in 6 vertices) are non-plannar.

A graph is plannar if and only if it contains no subgraphhomeomorphic to K

5or K

3,3.

A graph is plannar if and only if it contains no subgraphcontracible to K5 or K3,3.

Region or Face: Let G be a plannar graph. A region of G is the

area in which any arbitrary pair of points may be joined by a

curve without intersecting any edges of G.

Let G be a connected planar graph with n vertices, m edgesand r regions then n-m+r=2. This interrelation was

proposed by Euler.

Let G be a plannar graph with n vertices, m edges, r regionsand k components. Then n-m+r=k+1.

If G is a connected simple plannar graph with n3 verticesand m edges then m3n-6.

If in addition G has no triangle then m2n-4.


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/* Program for checking planarity of simple graph */

#include

#include

#define max 20

int adj[max][max],c=0; /*Adjacency matrix */

int n; /* Denotes number of nodes in the graph */

main()

{

int max_edges,i,origin,destin, no_edges;


scanf("%d",&n);

if(n


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if( (origin==0) && (destin==0) )

break;

if( origin > n || destin > n || origin


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OUTPUT

Test for the graph K3.

Enter number of vertices : 3Enter edge 1 (end vertices) : 1

2


3



3


2Enter edge 6 (end vertices) : 2

1

Graph is planar.


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OUTPUT

Test for the graph K5.

Enter number of vertices : 5Enter edges 1 (end vertices) : 1

2

Enter edges 2 (end vertices) : 1

3



5


1Enter edges 6 (end vertices) : 2

3


4



1


2


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4



1


2Enter edges 15 (end vertices) : 4

3


5


1


2


3


4

Graph is not planar.


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VERTEX COLOURING

Let G be a graph. The vertex colouring of G is labelling ofits vertices by some given colours such a way that no two

adjacent vertices have same colour.

K- Colourable Graph: Let G be a given graph. It is said to be K-

colourable if it can be coloured by k colours.

Chromatic Number for Vertex: If a graph G is k colourable but

not k-1 colourable then the chromatic number of G is k. The

chromatic number of a grapf G is denoted by (G).

If G is a simple graph with largest vertex degree , then G is(+1) colourable.

If G is a simple connected graph which is not a completegraph, and if the largest vertex degree of G is (3), then G

is - colourable.

Every simple planar graph is 6- colourable.Every simple planar graph is 5- colourable.Recent research (1976) have also proved that a planar graph

is four clolourable.

Vertex colouring has various uses in network problems.

Colouring of a map can be easily done with its help. For anygiven map, we can construct its dual graph as follows. Put a

vertex inside each region of the map and connect two distinct

vertices by an edge if and only if their respective regions share a

whole segment of their boundaries in common. Then, a proper

vertex coloring of the dual graph yields a proper coloring of theregions of the original map.


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GSM phone networks are constructed with the help of

vertex coloring. GSM is a cellular network with its entiregeographical range divided into hexagonal cells. Each cell has a

communication tower which connects with mobile phoneswithin the cell. All mobile phones connect to the GSM network

by searching for cells in the immediate vicinity. GSM networks

operate in only four different frequency ranges. The reason why

only four different frequencies suffice is clear: the map of the

cellular regions can be properly colored by using only four

different colors! So, the vertex coloring algorithm may be used

for assigning at most four different frequencies for any GSM

mobile phone network.


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APPLICATION OF GRAPH THEORY

Many applications of graph theory exist in the form

network analysis. These split broadly into three categories:

1. First, analysis to determine structural properties of anetwork, such as the distribution of vertex degrees and

diameter of the graph. A vast number of graph measures

exist, and the production of useful ones for various

domains remains an active area of research.

2.Second, analysis to find a measurable quantity within thenetwork, for example, for a transportation network, the

level of vehicular flow within any portion of it.

3. Third, analysis of dynamical properties of networks.Graph theory is also used to study molecules in chemistry

and physics. In condensed matter physics, the three dimensional

structure of complicated simulated atomic structures can be

studied quantitatively by gathering statistics on graph-theoretic

properties related to the topology of the atoms.

Graph theory is also widely used in sociology as a way, forexample, to measure actors prestige or to explore diffusion

mechanisms, notably through the use of social network analysis

software.

Likewise, graph theory is useful in biology and

conservation efforts where a vertex can represent regions where

certain species exist (or habitats) and the edges represent

migration paths, or movement between the regions. This


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information is important when looking at breeding patterns or

tracking the spread of disease, parasites or how changes to the

movement can affect other species.

Most important advances in graph theory arose as a result

of attempts to solve particular practical problems- Euler and the

bridges of Konigsberg, Caley and the enumeration of chemical

molecules, and Kirchhoffs work on electrical networks, etc.

There are also some attempts to solve on shortest path

problems. These problems are very interesting to discuss. There

are two problems Travelling salesman problem and Chinesepostman problem both are shortest path problems. A salesman

or a postman has to decide as to which path he should go so that

he would travel all the cities or villages and return to the starting

point in least possible total distance.

In order to solve these problem there are several

algorithms. Our main aim of this project is to present the C-

programming of these a few algorithms.

Dijkstras Technique

This technique is used to determine the shortest path

between two arbitrary vertices in a graph. Let w(vi,vj) is

associated with every edge (vi,vj) in a given graph G.

Furthermore, the weights are such that the total weight from

vertex vi to the vertex vk via vertex vj is w(vi,vj)+w(vj+vk). Using

this technique the weight from a vertex vs (starting of the path)

to vertex vt (the end of the path) in the graph G for a given path

(vs,v1), (v1,v2), (v2,v3), , (vi, vt) is given by w(vs,v1) + w(v1,v2) +


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w(v2,v3) +...+w(vi, vt). In a graph there are many possible paths

between vs and vt.

Dijkstras method is very popular and efficient one to

find every path from starting to terminal vertices. If there is an

edge between two vertices, then the weight of this edge is its

length. If several edges exist however, use the shortest edge. If

no edge actually exists set the length to infinity. Edge (vi,vj) does

not necessarily have the same length as edge (vj,vi). This allows

different paths depending on the direction of travel.

Dijkstras technique is based on assigning labels to each

vertex. The label is equal to the distance (weight) from the

starting vertex to that vertex. Obviously, the starting vertex vs

has a label 0. A label can be in one of two state-temporary or

permanent. A permanent label that lies along the shortest path

while a temporary label is one that has uncertainty as to label is

along the shortest path.


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Dijkstras Algorithm

Step 1: Assign a temporary label 1 (vs) = to all vertices

except vs.

Step 2: [Mark vs as permanent by assigning 0 label to it]

1 (vs)=0

Step 3: [Assignment value] of vs to vr where vr is last vertex to

be made permanent]

vr=vs

Step 4: If 1(vi)>1(vk)+w(vk,vi)

1(vi)=1(vk)+w(vk,vi)

Step 5: vr=vi

Step 6: If vt temporary label, repeat Step 4 to Step 5 otherwise

the value of vt is permanent label and is equal to the

shortest path vs to vt.

Step 7: Exit.


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/*Program of shortest path between two node in

graph using Djikstras algorithm */

#include

#define MAX 10

#define TEMP 0

#define PERM 1

#define infinity 9999

struct node

{

int predecessor;

int dist; /*minimum distance of node from source*/

int status;

};

create_graph();

int adj[MAX][MAX];

int n;

int findpath(int s,int d,int path[MAX],int *sdist)

{

struct node state[MAX];

int i,min,count=0,current,newdist,u,v;

*sdist=0;


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/* Make all nodes temporary */

for(i=1;i


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{

newdist=state[current].dist + adj[current][i];

/*Checks for Relabeling*/

if( newdist < state[i].dist )

{

state[i].predecessor = current;

state[i].dist = newdist;

}

}

}/*End of for*/

/*Search for temporary node with minimum distand make it

currentnode*/

min=infinity;

current=0;

for(i=1;i


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}

}/*End of for*/

if(current==0) /*If Source or Sink node is isolated*/

return 0;

state[current].status=PERM;

}/*End of while*/

/* Getting full path in array from destination to source */

while( current!=0 )

{

count++;

path[count]=current;

current=state[current].predecessor;

}

/*Getting distance from source to destination*/

for(i=count;i>1;i--)

{

u=path[i];

v=path[i-1];

*sdist+= adj[u][v];

}

return (count);


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}/*End of findpath()*/

void main()

{

int i;

int source,dest;

int path[MAX];

int shortdist,count;

create_graph();

while(1)

{

printf("Enter source vertex (end vertex) : ");

scanf("%d",&source);

printf("Enter destination vertex (end vertex) : ");

scanf("%d",&dest);

if(source==0 || dest==0)

break;

count = findpath(source,dest,path,&shortdist);

if(shortdist!=0)

{


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printf("Shortest distance is : %d\n", shortdist);

printf("Shortest Path is : ");

for(i=count;i>1;i--)

printf("%d->",path[i]);

printf("%d",path[i]);

printf("\n");

}

else

printf("There is no path from source to destination

node\n");

}/*End of while*/

}/*End of main()*/

create_graph()

{

int i,max_edges,origin,destin,wt;


scanf("%d",&n);

max_edges=n*(n-1);


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for(i=1;i n || destin > n || origin


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for(i=1;i


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OUTPUT

Shortest path between vertices 5 and 3 of followinggraph through c programming :

Enter number of vertices : 5


2



3



4



5



1


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4Enter weight for this edge : 10


1




Enter edge 9 (end verties) : 5

4



3



2



1




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2





2

Enter weight for this edge : 20Enter edge 16 (end vertices) : 1

2



2



2



2



2



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Enter source vertex : 5

Enter destination vertex : 3

Shortest distance is : 49Shortest path is : 5->4->1->2->3

Enter source vertex :


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BIBLIOGRAPHY

Gupta M. K., Discrete Mathematics, Edition 2008, chapter 7-11,

pages 327-502.Patel, R. B., Data structure through C, Edition 2008, chapter 8,

pages 591-664.

Wilson, R. J., Graph Theory ,Edition 2007, chapter 1-6 , page 1-

96.

Website:http://www.indiastudychannel.com/
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