Graph Theory

• What is Graph Theory??? • Who made this?• Polyhedra and networks• Applications of graphs• Definitions• Examples• Multigraph and Pseudograph • Weighted graph• Finite and Infinite Graph

Contents

What is that?!?

• Graph theory is the study of graphs

• A "graph" in this context refers to a collection of vertices or 'nodes' and a collection of edges that connect pairs of vertices.

• The graphs studied in graph theory should not be confused with "graphs of

functions"

• The paper written by Leonhard Euler on the Seven Bridges of Königsberg and published in 1736 is regarded as the first paper

in the

Who made this?

history of graph theory.

Seven Bridges of Königsberg

• The Königsberg Bridge problem

Polygon

• A closed figure bounded by straight line segments is called a polygon.

• A polygon having sides of equal length and equal angles is called a regular polygon.

Polyhedra and networks

• A polyhedron is often defined as a geometric solid with flat faces and straight edges.

• It is a 3D solid whose faces are polygons joined at their edges.

• A polyhedron is said to be regular if its faces are made up of regular polygons.

• The Greek word ‘poly’ (many) and Indo-European word ‘hedron’ (seat).

Regular polyhedron

Applications of graphs

• Engineering

• Physical, Social and Biological Sciences

• Utilities problem

• Electrical Network problems

• Seating problem

Definitions

• Graph : a graph G is a collection of points or nodes called ‘vertices’ and a collection of lines called ‘edges’ each of which joins either a pair of vertices or a single vertex to itself.

• A graph G is an ordered pair (V,E), where V is non-empty set of vertices and E is the mon-empty set of edges.

• It is denoted by G = (V,E). If v1 and v2 are two vertices of G, the edge joining v1 and v2 is denoted by either v1v2 or v2v1.

Example 1

• Let G = (V,E) be a graph, where V={a,b,c,d,e}, E={ab,bc,cd,de} = {e1,e2,e3,e4}. This graph is shown as …….

Note

• Suppose e is an edge joining the vertices u and v, then the vertex u and edge e are said to be incident with each other.

• If two distinct edges are incident with a common vertex, then they are called adjacent edges.

Example-2• Consider the graph G = (V,E) , where

V={v1,v2,v3,v4,v5} , E= {e1,e2,e3,e4,e5,e6 }.

• e1 is incident with v1 and v2 or we can say that the vertices v1 and v2 are incident with the edge e1.

Also the edge e1

is adjacent with

e2 and with e3

but not with e5.

• Graph with m number of vertices and n number of edges is called a (m,n) graph.

• Edge joining a vertex to itself is called a loop or a self – loop.

Example-3

• The edge e2 is a self-loop. If there are more than one edges joining a pair of vertices, then these edges are called multiple edges or parallel edges .In the figure e4 and e5 are parallel edges. A graph without loops and multiple edges is called a simple graph .

Multigraph

• A Multigraph is a graph which has multiple edges, (also called "parallel edges“), that is, edges that have the same end nodes.

• Thus two vertices may be connected by more than one edge. Formally, a Multigraph G is an ordered pair G =(V, E) with

1. V a set of vertices or nodes,

2. E a multiset of unordered pairs of vertices, called edges or lines

Multigraph

Pseudograph

• Pseudograph :

If both multiple edges and loops are permitted, then it is called a Pseudograph.

Weighted graph

• If in a graph G, each edge or each vertex are associated with a positive real number, then the graph is called a weighted graph.

• For example, in a graph showing the system of water pipelines ,each pipeline maybe associated with the amount of water that flows in the pipe.

Weighted graph

Finite and Infinite Graph

• A graph with finite number of edges is called a finite graph.

• A graph with infinite number of edges is called a infinite graph.

Finite and Infinite Graph

This is an infinite graph

Degree of vertex

• The degree of vertex is the number of edges incident on a vertex v with self-loops counted twice.

• It is denoted by d(v).

Degree of vertex

• The degrees of U,V,W and X are respectively 2,2,3 and 3 respectively.

• The degree of X is 3 because a self loop has a degree of 2.

[Isolated (and) pendent] vertex

• A vertex having no incident edges is called an isolated vertex. The degree of an isolated vertex is zero.

• A vertex of degree ‘one’ is called a pendent vertex.

[Isolated (and) pendent] vertex

In the given graph, E is a pendent vertex and F is an isolated vertex.

Null graph

• If every vertex of a graph is an isolated vertex, then it is called a null graph.

Even and odd degree vertices

• A vertex is called an even degree vertex if its degree is even.

• A vertex is called an odd degree vertex if its degree is odd.