This PowerPoint contains basic introduction of graph theory

1. GRAPH THEORY
Mr. S.L. Khairnar

2. Graph Theory

4. A graph is an ordered pair G = (V, E), where V is the set of
vertices and E is the set of edges which connects the
vertices of V
A graph is represented with dots or circles (vertices) joined by lines (edges)
The magnitude of graph G is characterized by number of vertices |V| (called the order of G)
and number of edges |E| (size of G)

6. Directed Graph

8. An edge e  E of a directed graph is represented as an ordered pair (u,v), where u, v 
V. Here u is the initial vertex and v is the terminal vertex. Also assume here that u ≠ v
2
4
3
1
V = { 1, 2, 3, 4}, | V | = 4
E = {(1,2), (2,3), (2,4), (4,1), (4,2)}, | E |=5

9. Undirected Graph

11. 2
4
3
1
V = { 1, 2, 3, 4}, | V | = 4
E = {(1,2), (2,3), (2,4), (4,1)}, | E |=4
An edge e  E of an undirected graph is represented as an
unordered pair (u,v)=(v,u), where u, v  V. Also assume that u ≠ v

12. Degree of a Vertex

14. Degree of a vertex in an undirected graph is the number of edges incident on
it. In a directed graph, the out degree of a vertex is the number of edges
leaving it and the in degree is the number of edges entering it
2
4
31
2
4
3
1
The degree of vertex 2 is 3 The in degree of vertex 2 is 2
and the in degree of vertex 4
is 1

15. Note. A vertex of degree zero is called isolated.
Note. A vertex is pendant if and only if it has degree one

16. Theorem
The Handshaking Theorem

17. Let G = (V; E) be an undirected graph with m
edges. Then 2m = 𝑣 ∈𝑉 deg 𝑣

18. Theorem
An undirected graph has an even
number of vertices of odd
degree.

19. Example
How many edges are present in a
graph with degrees as
1,2,3,4,5,6,7

21. In Degree and Out
Degree of a Vertex
14/08/2020

22. In a graph with directed edges
the in-degree of a vertex v,
denoted by deg−
𝑣, is the number
of edges with v as their terminal
vertex

23. In a graph with directed edges
the Out-degree of a vertex v,
denoted by deg+
𝑣, is the number
of edges with v as their initial
vertex

24. Theorem
The Handshaking Theorem

25. Let G = (V; E) be an undirected graph with m
edges. Then
m = 𝑣 ∈𝑉 deg+
𝑣= 𝑣 ∈𝑉 deg−
𝑣