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Lecture 09: Basic Graph Theory

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JC Liu. MACM101 Discrete Mathematics I. 1. Lecture 09: Basic Graph Theory. Introduction to Graphs ... The two graphs below are in fact the same graph (structure-wise) ... – PowerPoint PPT presentation

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Title: Lecture 09: Basic Graph Theory


1
Lecture 09 Basic Graph Theory
  • Introduction to Graphs
  • Basic Definitions
  • Complete Graph/Cycle
  • Bipartite Graphs
  • Operations on Graphs
  • Other Graph Representations
  • Graph Isomorphism

2
9.1. Basic Definitions
  • A graph G (V, E)
  • Is a discrete structure consisting of
  • A nonempty set of vertices (nodes) V and
  • a set of edges E that connect these vertices.
  • First consider graph that contains
  • No arrows (directions)
  • No loops edges from a vertex to itself.
  • No multiple edges joining the same two vertices.
  • Also referred to as undirected simple graph.

3
9.1. Basic Definitions
  • A graph can be described either by the use of
  • Sets or
  • a diagram.
  • The diagram can be drawn differently and still
    represents the same graph.
  • Example
  • Sets A graph with vertices V ? A, B, C, D ?
    and edges E ? ?A,B?, ?B,C?, ?C,D? ?.

4
9.1. Basic Definitions
  • Adjacent, incident, and degree
  • Two vertices u and v in V are adjacent or
    neighbors if there is an edge e between u and v.
  • The edge e ?u, v? joins (or connects/links) the
    vertices u and v.
  • The vertices u and v are endpoints of e.
  • The edge e is incident with the vertex u and v.
  • The degree of u, denoted deg(u), is the number of
    edges incident with a vertex u.

5
9.1. Basic Definitions
  • Example
  • Vertices A and B are adjacent.
  • Vertices A and D are not because there is no edge
    between them.
  • deg(A) 1
  • deg(B) 3
  • deg(C) 0, C is called isolated.

6
9.1. Basic Definitions
  • The Handshaking Theorem
  • Let G (V, E). Then
  • Proof
  • Each edge contributes twice to the degree count
    of all vertices. (Q.E.D.)
  • Example
  • If a graph has 3 vertices, can each vertex have
    degree 3? Or 4?
  • The sum is 3?3 9 which is an odd number. Not
    possible.
  • The sum is 4?3 12 2 E and 12/2 6 edges.
    May be possible.

7
9.2. Some Special Graphs
  • Complete graphs - Kn
  • The (simple) graph with
  • n vertices
  • exactly one edge between every pair of distinct
    vertices.
  • Examples
  • K5 is important because it is the simplest
    nonplanar graph. It cannot be drawn in a plane
    with nonintersecting edges.

8
9.2. Some Special Graphs
  • Complete graphs - Kn
  • How many edges of Kn?

9
9.2. Some Special Graphs
  • Cycles Cn
  • Is an n vertex graph with all vertices having a
    degree two, and all are connected.
  • Examples

10
9.3. Bipartite Graphs
  • A bipartite graph G is
  • Is a graph in which the vertices V can be
    partitioned into two disjoint subsets V1 and V2
    such that
  • No two vertices in V1 are adjacent.
  • No two vertices in V2 are adjacent.

11
9.3. Bipartite Graphs
  • Examples
  • Let the courses and professors be the vertices of
    a graph G.
  • Put an edge between a course and a professor
    whenever the professor can teach the course.
  • The resulting graph is bipartite.

12
9.3. Bipartite Graphs
  • Examples
  • Is a Star network (K1,n) bipartite graph?
  • Is a cycle Ck bipartite graph?

13
9.3. Bipartite Graphs
  • Examples
  • Is the following graph bipartite?
  • Solution
  • If a is in V1 then b, d and e must be in V2
    (why?)
  • Then c is in V1 and there is no inconsistency.
  • The graph is a complete bipartite graph K2,3.

14
9.3. Bipartite Graphs
  • A bipartite graph G is
  • Is a graph in which the vertices V can be
    partitioned into two disjoint subsets V1 and V2
    such that
  • No two vertices in V1 are adjacent.
  • No two vertices in V2 are adjacent.
  • A complete bipartite graph Km,n is
  • Is a bipartite graph in which the sets V1 and V2
    contain m and n vertices, respectively, and every
    vertex in V1 is adjacent to every vertex in V2.
  • How many edges ?

15
9.4. Operations on Graphs
  • A subgraph of G (V, E)
  • Is a graph H (W, F) where W ? V and F ? E.
  • Examples
  • K5 and its subgraph

16
9.4. Operations on Graphs
  • The union of two graphs G1 (V1, E1) and G2
    (V2, E2)
  • Is a graph with vertex set V1 ? V2 and edge set
    E1 ? E2.
  • i.e. G1 ? G2 (V1 ? V2, E1 ? E2)
  • Examples
  • G1 and G2 with their union G1 ? G2

17
9.5. Other Graph Representations
  • To analyze a graph using a computer, we need to
    communicate the vertices and edges of the graph.
  • Graph representations
  • Adjacency list
  • Adjacency matrices

18
9.5. Other Graph Representations
  • Adjacency list of a graph G
  • Specify the vertices that are adjacent to each
    vertex of the graph.
  • Examples
  • A graph and its adjacency list.

19
9.5. Other Graph Representations
  • Adjacency matrix of a graph G (V, E) with n
    vertices
  • Is an n ? n matrix A aij, where
  • Sum of the entries in row i of the adjacency
    matrix ?
  • The matrix is symmetric, i.e. aij aji.

20
9.5. Other Graph Representations
  • Examples
  • Two graphs and their adjacency matrices.

21
9.6. Graph Isomorphism
  • The two graphs below are in fact the same graph
    (structure-wise).
  • We say that these graphs are isomorphic.

22
9.6. Graph Isomorphism
  • Let G1 (V1, E1) and G2 (V2, E2) be simple
    graphs.
  • The graphs G1 and G2 are isomorphic iff
  • There exists a bijection f V1?V2
  • The function f is called an isomorphism of G1
    with G2.
  • i.e. For all vertices u and v in V1,
  • if u and v are adjacent in G1
  • then f(u) and f(v) are adjacent in G2.

23
9.6. Graph Isomorphism
  • Graph isomorphism invariant properties
  • Are the properties that G1 and G2 must have in
    common in order to be isomorphic
  • the same number of vertices.
  • the same number of edges.
  • degrees of corresponding vertices are the same.
  • if one is bipartite, the other must be.
  • if one is complete, the other must be.
  • etc. but these are necessary, not sufficient !

24
9.6. Graph Isomorphism
  • Examples
  • The following graphs are isomorphic.

25
9.6. Graph Isomorphism
  • Examples
  • The following graphs are not isomorphic. Why?

26
9.6. Graph Isomorphism
  • Examples
  • The following graphs are not isomorphic.
  • Vertex C in G1 is adjacent to the vertices A, B,
    D and E. So it has a degree of 4.
  • But there are no vertices of degree 4 in G2.

27
9.6. Graph Isomorphism
  • Examples
  • Determine if the following two graphs G1 and G2
    are isomorphic.

28
9.6. Graph Isomorphism
  • Solution
  • Check . . .
  • They have the same number of vertices 5
  • They have the same number of edges 8
  • They have the same number of vertices with the
    same degrees 2, 3, 3, 4, 4.

29
9.6. Graph Isomorphism
  • 3?2, 1?1, 5?4, 2?3, 4?5
  • The problem in general is very difficult, even
    using
  • a computer !

30
9.7. Further Readings
  • Introduction to Graphs
  • Basic Definitions Section 11.1
  • Bipartite Graphs p.541
  • Graph Isomorphism Section 11.2
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