Ali Mahmudi, Introduction to Graph Theory [8]
D. Some Type of Graphs
Note that in the definition of graph, we do not exclude the possibility that the two endpoints of an edge are the same vertex. This called loop, for obvious
reasons. Also, we may have multiple edges, which is when more than one edge shares the same set of endpoints, i.e, the edges of the graph are not uniquely
determined by their endpoints. In another word, two more edges that have end vertices are called multiple edges or parallel edges.
A vertex of a graph which is not the end of any edge is called isolated. Two vertices which are joined by an edge are said to be adjacent or neighbors. The set
of all neighbors of a fixed vertex v of G is called the neighborhood set of v and is denoted by Nv. Given graphs G
= VG, EG, where VG = {v
1
, v
2
, v
3
, v
4
} and EG
1
= {v
1
v
2
, v
1
v
3
, v
3
v
4
}. We can graphically represent this graph in Figure 1.8 below.
Figure 1.8. Graph with multi edges, loop, and isolated vertex In the graph in Figure 1.8, e
2
and e
3
are multiple-edges, edge e
5
is loop, and E is an isolated vertex. The neighborhood of vertex B, D, and E, are NB = {A,
C}, ND = {C}, and NE = { } respectively.
1. Simple Graph and Multi-Graph
E
D C
B A
e
2
e
3
e
1
e
5
e
4
Definition 1.2
A graph is called simple if it has no loops and no multi-edges. In a simple graph, we often identify an edge using its endpoints, e.g, edge u, v.
A graph having multiple edges but no loop is called a multi-graph.
Ali Mahmudi, Introduction to Graph Theory [9]
2. Complete Graph
If the complete graph has vertices v
1
, v
2
, …, v
n
then the edge set can be given by
E = {v
i
, v
j
; v
i
v
j
; i,j = 1, 2, …, n} Note that the grah has
1 n
n 2
1 edges, since there are n – 1 edges incident
with each of the n vertices v
i
, so a total of nn – 1, but divide by 2, since v
i
, v
j
= v
j
, v
i
. A complete graph with n vertices is denoted by K
n
. Figure 1.9 shows K
1
, K
2
, …, K
5
.
Figure 1.9 The Complete Graphs on at most five vertices
3. Empty Graph
K
1
K
2
K
4
K
5
K
3
Definition 1.3
A complete graph is a simple graph in which each pair of distinct vertices is joined by an edge.
Definition 1.4
An empty null graph is a graph with no edges.
Ali Mahmudi, Introduction to Graph Theory [10]
An empty graph with n vertices is denoted by N
n
. Figure 1.10 shows N
1
, N
2
, …, N
5
.
Figure 1.10 The Null Graphs on at most five vertices
4. Bipartite Graph
Figure 1.11 shows bipartite graphs G with partition X = {a, c, e} and Y = {b, d, f}.
Figure 1.11 Bipartite Graph G
a
b c
d e
f G
e
f
a c
b
d
N
1
N
2
N
3
N
4
N
5
Definition 1.5
Let G be a graph. If the vertex set VG can be partitioned into two nonempty subsets X and Y, i.e. X
Y = VG and X Y = , ins such a way that each edge of G has one end in X and Y then G is called bipartite
graph. Notice that there is no edge connecting two vertices in the same subset.
Represent
Definition 1.6
A complete bipartite graph is a simple bipartite graph G, with partition VG = X
Y, in such every vertex in X is joined to every vertex of Y. If X has m vertices and Y has n vertices, such a graph is denoted K
m,n
.
Ali Mahmudi, Introduction to Graph Theory [11]
Consider that since each of the m vertices in the partition set X of K
m,n
is adjacent to each of the n vertices in the partition set Y, K
m,n
has m x n edges. Figure 1.12 shows complete bipartite graphs K
2,3
and K
3,3
.
Figure 1.12 Complete Bipartite Graph
Proof
Suppose a graph contains a triangle {u, v, s}. By definition, we have edges uv, vs, and su.
We prove the graph is not bipartite by contradiction. Assume that it is bipartite. Then we can have a partition {U, V} of vertex set.
Consider which of the two subsets U, V each of the three vertices u, v, s belongs. Clearly, there must be at least two vertices among u, v, s belonging to the same
subset, which means there is no edge connecting them. Contradiction.
E.
Vertex Degrees
K
2,3
K
3,3
Theorem 1.1
A graph is not bipartite if it contains a triangle. A triangle in a graph is three vertices such that any two of them are neighbors.
Denition 1.5
Let v be a vertex of the graph G. The degree of v, denoted by dv or d
G
v, is the number of edges of G incident with v, counting each loop twice, i.e, it is
the number of times v is an end vertex of an edges.
Ali Mahmudi, Introduction to Graph Theory [12]
Minimum and maximum degree of G, denoted G dan G respectively,
are defined below. G = minimum {dv | v VG}
G = maxsimum {dv | v VG}
Consider Figure 1.13 below.
Figure 1.13 In the graph of Figure 1.13 we have da = 1, db = 3, dc = 4, dd = 3,
de = 1, and df = 0. We also have
G = 0 and
G = 4.
Note that da + db + dc + dd + de + df = 12 = 2 x the number of edges of G.
Proof
Each edge, since it has two end vertices, contributes precisely 2 to the sum of degrees, i.e., when the degrees of the vertices are summed each edge is counted
twice. Note that even a loop contributes 2 although the 2 ends are identical.
Notice that in the Figure 1.13 there is an even number of odd vertices. A vertex of a graph is called odd or even depending on whether its degree is odd or
even. f
a b
c d
e
G
Theorem 1.2 Handshaking Theorem
For any graph G |
G E
| 2
v d
G V
v
where |EG| = the number of edges in the graph.
Ali Mahmudi, Introduction to Graph Theory [13]
Proof
Let A is the set of odd vertices of G and B is the set of even vertices of G. Then, for each a
A, da is even and so
A a
a d
, being a sum of even numbers, is even. By Theorem 1.2, we get
A a
a d
+
B b
b d
=
G
V v
v d
= 2|EG| Thus
B b
b d
= 2|EG| –
A a
a d
is even being the difference of two even numbers. As all the terms in
B b
b d
are odd and their sum is even there must be an even number of them because the sum of an odd number of odd number is odd.
Corollary 1.3 In any graph G there is an even number of odd vertices
Theorem 1.3
For every graph G, G
| |
| |
2 G
V G
E G,
where |VG| = number of vertices of G.
Definition 1.5
If for some positive integer k, dv = k for every vertex v of the graph G, then G is called k-regular.
Ali Mahmudi, Introduction to Graph Theory [14]
A complete graph is one that is k-regular for some k. The graph drawn below graph G is a 2-regular and graph K
4
is a 3-regular.
Figure 1.14 Regular.Graph Notice that the complete graph K
n
is n-1-regular. The complete bipartite graph K
n,n
is n-regular.
F. Isomorphic Graphs
Kategori