AN INTRODUCTION TO GRAPH THEORY PRESENTED BY DR

AN INTRODUCTION TO GRAPH THEORY PRESENTED BY , DR. N. NEELA AP / MATHEMATICS KIOT , SALEM

INTRODUCTION The first paper in graph theory was written by Euler in 1736 when he settled the famous unsolved problem of his day , known as the Konigsberg Bridge problem. Konigsberg is now called Kaliningrad and is in Lithuania. Two islands A and B and the two banks C and D of the Pragel river are connected by seven bridges.

CONTD The problem was to start from any one of the land areas , walk across each bridge exactly once and return to the starting point. Euler proved that this problem has no solution.

CONTD � Euler abstracted the problem by replacing each land area by a point and each bridge by a line joining these points leading to a Multigraph.

GRAPH A Graph G consists of a pair (V(G) , X(G)) where V(G) is a non empty finite set whose elements are called points or vertices and X(G) is a set of un ordered pairs of distinct elements of V(G). � The elements of X(G) are called lines or edges. �

ADJACENT � If x={u, v} in X(G) , the line x is said to join u and v. we say that the points u and v are adjacent. We also say that the point u and the line x are incident with each other. If two distinct lines x and y are incident with a common point then they are called adjacent lines.

MULTIGRAPH If more than one line joining two vertices are allowed , the resulting graph is called multigraph. � Lines joining the same points are called multiple lines. � If further loops are also allowed , the resulting graph is called pseudo graph. �

COMPLETE GRAPH A graph in which any two distinct points are adjacent is called a complete graph. � Triangle is a complete graph with 3 vertices. �

NULL GRAPH A graph whose edge set is zero is called a null graph. � A graph is called labelled if its points are distinguished from one another by names such as 1, 2, 3, 4 , …. �

BIGRAPH � A graph is called a Bigraph or Bipartite graph if V(G) can be partitioned into two disjoint subsets V 1 and V 2 such that every line of G joints a point of V 1 to a point of V 2.

DEGREE The degree of a point is the number of lines incident with that point. � A point with degree 0 is called isolated point. � A point with degree 1 is called end point. �

REGULAR GRAPH � If all the points of a graph have the same degree r , then the graph is called regular graph of degree r.

WALK , TRAIL , PATH A Walk of a graph is an alternating sequence of points and lines beginning and ending with points. � A Walk is called trail if all its lines are distinct. � A Walk is called path if all its points are distinct. � A walk is called closed If the initial point and the end points are same. �

CYCLE �A closed walk in which [the number of vertices are greater than or equal to 3 ] the initial and end points are same and the remaining points are distinct is called a cycle. � Triangle is a cycle with 3 points.

CONNECTED GRAPH A graph is said to be connected if every pair of its points are connected. � A graph which is not connected is said to be disconnected. �

EULERIAN GRAPH � A closed trail containing all points and lines is called an Eulerian trail. � A graph having an Eulerian trail is called an Eulerian graph.

TREE A graph that contains no cycles is called an acyclic graph. � A connected acyclic graph is called a tree. �

COLOURING � An assignment of colours to the vertices of a graph so that no two adjacent vertices get the same colour is called a colouring of a graph.

WEIGHTED GRAPH �A graph is called weighted if there is a real number associated with each edge. The real number associated with each edge is called its weight.

SUB GRAPH �A graph H= (V 1, X 1) is called a subgraph of G if V 1 is a subset of V and X 1 is subset of X. Then we say that H is a subgraph of G and G is a super graph of H.

APPLICATIONS � Fast Communication in sensor networks Using Radio Labeling. � Designing Fault Tolerant Systems with Facility Graphs. � Automatic channel allocation for small wireless local area network. � Avoiding Stealth Worms by Using Vertex Covering Algorithm � Graph Labeling in Communication Relevant to Adhoc Networks. � Reducing the Complexity of Algorithms in Compression Networks.