# 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

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

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

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

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

http://arief.ismy.web.id

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