Introduction To Graph Theory: Konigsberg's Bridge Problem

GRAPH THEORY:

INTRODUCTION TO GRAPH:

Graph is a pictorial representation of a physical situation involving discrete objects.
Graph Theory originally started with puzzles. A famous topologist from 17th century Leonhard Euler wrote the first paper ever in graph theory, because of its simplicity graph theory is used in various fields: Engineering,Physical,Social,and other areas.

The First Graph theory problem were the :
Konigsberg's Bridge Problem
Utilities Problem
CONCEPTS IN GRAPH:

DEFINITION :
A graph is a structure which is represented by G=(V,E) where,'V' is a set of discrete objects called vertices and 'E' is a set of edges which represents the relationship between the objects.

Consider the graph below:

FIGURE 1

G=( {V1,V2,V3,V4,V5

TRIVIAL GRAPH

A trivial graph G is a graph with only one vertex.

DIRECTED AND NON-DIRECTED GRAPH

In a directed graph each edge has a direction whereas in a non-directed graph the graph contains edges but the edges are not directed.

CONNECTED AND DISCONNECTED GRAPH

A graph G is said to be a connected graph if there exist a path between every pair of vertices There should be at least one edge for every vertex in the graph. and A graph G is said to a disconnected graph if there exist no path between every pair of vertices.

COMPLETE GRAPH

A simple graph with ‘n’ mutual vertices is called a complete graph and it is denoted by ‘Kn’. In the graph, a vertex should have edges with all other vertices, then it called a complete graph.

ISOMORPHISM OF GRAPHS

Two graphs G1 and G2 are said to be isomorphic when there is a bijection 'a' from the vertex set of G1 to the vertex set of G2 such that {a(x),a(y)} is an edge of G2 if and only if {x,y} is an edge of G1.The bijection a is said to be an isomorphic graph.
REPRESENTATION OF GRAPH –

FIGURE 2

TREES