Discrete Structures Chapter 7A

1
Discrete StructuresChapter 7A Graphs
Nurul Amelina Nasharuddin Multimedia Department
2
Objectives

• On completion of this topic, student should be
  able to
• Explain basic terminology of a graph
• Identify Euler and Hamiltonian cycle
• Represent graphs using adjacency matrices

3
Introduction to Graphs

• A graph G, consists of
• V, a nonempty set of vertices (or nodes) and
• E, a set of edges
• G (V, E)
• Each edge has either one or two vertices
  associated with it, called its endpoints. An edge
  is said to connect its endpoints
• Edge-endpoint function is the correspondence from
  edges to endpoints

4
Terminology

• Loop, parallel edges, isolated, adjacent,
  incident
• Loop - An edge connects a vertex to itself
• Two edges connect the same pair of vertices are
  said to be parallel
• Isolated vertex - Unconnected vertex
• Two vertices that are connected by an edge are
  called adjacent
• An edge is said to be incident on each of its end
  points

5
Example

• Vertex set u1, u2, u3
• Edge set e1, e2, e3, e4
• e1, e2, e3 are incident on u1
• u2 and u3 are adjacent to u1
• e4 is a loop
• e2 and e3 are parallel

6
Types of Graph

• Directed order counts when discussing edges
• Undirected (bidirectional)
• Weighted each edge has a value associated with
  it
• Unweighted

7
Examples of Graphs
8
Special Graphs

• Simple does not have any loops or parallel
  edges
• Complete graphs there is an edge between
  every possible tuple of vertices
• Bipartite graph V can be partitioned into V1
  and V2, such that
• (x,y)?E ? (x?V1 ? y?V2) ? (x?V2 ? y?V1)
• Sub graphs - G1 is a subset of G2 iff
• Every vertex in G1 is in G2
• Every edge in G1 is in G2
• Connected graph can get from any vertex to
  another via edges in the graph

9
Simple Graph

• Eg Draw all simple graphs with 4 vertices u, v,
  w, x and two edges, one of which is u, v

10
Complete Graph

• There is an edge between every possible tuple
  of vertices. e C(n,2) n.(n-1)/2

11
Bipartite Graph

• A graph is bipartite if its vertices can be
  partitioned into two disjoint subsets U and V
  such that each edge connects a vertex from U to
  one from V.

12
Complete Bipartite

• A bipartite graph is a complete bipartite graph
  if every vertex in U is connected to every vertex
  in V
• If U has m elements and V has n, then we denote
  the resulting complete bipartite graph by Km,n.
  The illustration shows K3,2

13
Degree of Vertex

• Defined as the number of edges attached
  (incident) to the vertex
• A loop is counted twice

14
Example

• Find the degree of each vertex and the total
  degree of graph G where the graph
• Contains 3 vertex v1, v2, v3
• Contains 3 edges e1, e2, e3
• Endpoints of e1 are v2 and v3
• Endpoints of e2 are v2 and v3
• e3 is a loop at v3
• v1 is an isolated vertex
• Ans Total degree 6

15
Handshake Theorem

• If G is ANY graph, then the sum of the degrees of
  all the vertices of G equals twice the number of
  edges of G
• Specifically, if the vertices of G are v1, v2, ,
  vn, where n is a nonnegative integer, then
• The total degree of G
• deg(v1)deg(v2)deg(vn)
• 2 ? (the number of edges of G)

16
Total Degree of a Graph is Even

• Prove that the total of the degrees of all
  vertices in a graph is even
• Since the total degree equals 2 times of edges,
  which is an integer, the sum of all degree is
  even.

17
Whether Certain Graphs Exist

• Draw a graph with the specified properties or
  show that no such graph exists
• (a) A graph with four vertices of degrees 1,1,2,
  and 3
• No such graph is possible. By Handshake Theorem
  the total degree is even. 1 1 2 3 7 not
  even
• (b) A graph with four vertices of degrees 1, 1,
  3 and 3
• (c) A simple graph with four vertices of degrees
  1, 1, 3 and 3

18
Even No. of Vertices with Odd Degree

• In any graph, there are an even number of
  vertices with odd degree
• Is there a graph with ten vertices of degrees
  1,1,2,2,2,3,4,4,4, and 6?
• Ans No. such a graph will have 3 vertices of
  odd degree (1,1,3) which is impossible.