Graph Theory Part 3

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					                                                                               Adjacency Matrices
                                                                      Two type of matrices:
                                                                      - Based on the adjacency of vertices
        Graph Theory                                                  - Based on incidence of vertices and edges
8.5. Representation of Graphs
                                                                      Definition
                                                                      - Suppose that G = (V, E) is a simple graph where
                                                                        |V| = n. Suppose that the vertices of G are v1, v2,
           Muhammad Arief                                               …, vn The adjacency matrix A of G (or AG) is the
  download dari http://arief.ismy.web.id                                n x n zero-one matrix with 1 as its (i, j)th entry
                                                                        when vi and vj, are adjacent and 0 as its (i, j)th
                                                                        entry when they are not adjacent .
                                           http://arief.ismy.web.id                                               http://arief.ismy.web.id




  Adjacency Matrices                                                                     Example
                                                                                     Adjacency Matrix?
               A = [aij]

 aij = 1    if {vi, vj} is an edge of G
     =0     otherwise




                                           http://arief.ismy.web.id                                               http://arief.ismy.web.id




             Example                                                                     Example
             The Graph?                                                              Adjacency Matrix?




                                           http://arief.ismy.web.id                                               http://arief.ismy.web.id
            Incidence Matrices                                                            Example
Definition
- Suppose that G = (V, E) is a simple graph where                                      Incidence Matrix?
  |V| = n. Suppose that the vertices of G are v1, v2,
  …, vn and the edges of G are e1, e2, …, em. The
  incidence matrix is the n x m matrix

                  M = [mij], where:
      mij   =1    when edge ej is incidence with vi
            =0    otherwise


                                            http://arief.ismy.web.id                                                     http://arief.ismy.web.id




                   Example

                 Incidence Matrix?
                                                                                      Graph Theory
                                                                              8.6. Isomorphisms of Graphs


                                                                                       Muhammad Arief
                                                                                download dari http://arief.ismy.web.id



                                            http://arief.ismy.web.id                                                     http://arief.ismy.web.id




       One-to-One Function                                                          Onto Function
Definition:                                                            Definition:
A function f is said to be one-to-one, or                              A function f from A to B is called onto, or
  injective, if and only if f(a) = f(b) implies that                     surjective, if and only if for every elements
  a = b for all a and b in the domain of f. A                            b ∈ B there is an element a ∈ A with f(a) =
  function is said to be an injection if it is one-
                                                                         b. A function f is called a surjection if it is
  to-one.
                                                                         onto.
Notation:
            ∀a∀b (f(a) = f(b) → a = b)                                 Notation:
            ∀a∀b (a ≠ b → f(a) ≠ f(b))                                             ∀y∃x ( f(x) = y)

                                            http://arief.ismy.web.id                                                     http://arief.ismy.web.id
One-to-One Correspondence                                               One-to-One Correspondence
Definition:
The function f is a one-to-one
 correspondence, or a bijection, if it is both
 one-to-one and onto.

A = {a, b, c, d}
B = {1, 2, 3, 4}
f(a) = 4, f(b) = 2, f(c) = 1, f(d) = 3.                                                One-to-one, onto
Is this a bijection?
Yes, because it is both one-to-one and onto.
                                             http://arief.ismy.web.id                                               http://arief.ismy.web.id




One-to-One Correspondence
                                                                              Isomorphisme of Graphs
                                                                        Definition
                                                                        - The simple graphs G1 = (V1, E1) and G2 = (V2,
                                                                          E2) are isomorphic if there is a one-to-one and
                                                                          onto function f from V1 to V2 with the property
                                                                          that a and b are adjacent in G1 if and only if f(a)
                                                                          and f(b) are adjacent in G2, for all a and b in V1.
                                                                          Such a function f is called an isomorphism.
              One-to-one, not onto
              Onto, not one-to-one
              One-to-one, onto
              Neither one-to-one, nor onto
              Not a function
                                             http://arief.ismy.web.id                                               http://arief.ismy.web.id




Example                                                                                    Example
 Show that G = (V, E) and                                               The function f with f(u1) = v1, f(u2) = v4, f(u3) = v3
   H = (W, F) are                                                         and f(u4) = v2 is a one-to-one correspondence
   isomorphic                                                             between G and H. For every adjacent vertices in
                                                                          G, there is also corresponding adjacent vertices in
                                                                          H.




                                             http://arief.ismy.web.id                                               http://arief.ismy.web.id
            What to check?                                                           Example
 Have the same number of vertices.                                  Determine whether the graphs shown below are
                                                                     isomorphic.
 Have the same number of edges OR
 have the same number of the degree of the
   vertices.
 Have the same number of the degree of the
   adjacent vertices.
 Have the same adjacency matrices.




                                       http://arief.ismy.web.id                                            http://arief.ismy.web.id




                 Example                                                             Example
Determine whether the graphs shown below are                      Determine whether the graphs shown below are
 isomorphic.                                                       isomorphic.




                                       http://arief.ismy.web.id                                            http://arief.ismy.web.id




                 Example




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