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
It is often the case that two graphs have the same structure, differing only in the way their vertices and edges are labeled or only in the way they are
represented geometrically. For many purposes, we can regard the two graphs as essentially the same. This essential likeness has a special name and we now define
this formally.
In other word, the graphs G
1
and G
2
are isomorphic if the vertices of G
1
can be paired off with the vertices of G
2
and the edges of G
1
can be paired off with the edges of G
2
in such way that the ends paired off edges are paired off. Thus G
1
is really just the same graph as G
2
, apart from a possible change in how the vertices 2-regular
3-regular
Definition 1.6
A graph G
1
= V
1
, E
1
is said to be isomorphic to the graph G
2
= V
2
, E
2
if there is one-to one correspondence between the edge sets E
1
and E
2
in such a way that if e
1
is an edge with end vertices u
1
and v
1
in G
1
then the corresponding edge e
2
in G
2
has its end points the vertices u
2
and v
2
in G
2
which correspond to u
1
and v
1
respectively. Such a pair of correspondence is called a graph isomorphism.
Ali Mahmudi, Introduction to Graph Theory [15]
and edges are named or a possible redrawing of the graphs. Figure 1.8 shows five fairly obvious pairs of isomorphic graphs.
a
b
Figure 1.15 Isomorphic Graphs Figure 1.15 gives some examples to illustrate that often it can be quite
difficult to determine if two graphs are isomorphic. In the Figure 1.15 a an isomorphism is given by the following one-to-one correspondence of vertices:
A P, B Q, C R, D S
Note how A is joined with B and C, but not to D and similarly P is joined to Q and R, but not to S.
The problem of determining when two graphs are isomorphic gets harder as the number of vertices and edges of the graphs get larger. Clearly, if two graphs
G
1
and G
2
are isomorphic then they must have i
The same number of vertices ii
The same number of edges A
B C
D P
Q R
S
Ali Mahmudi, Introduction to Graph Theory [16]
However, as we will now see, both conditions are not sufficient. The graph G of Figure 1.16 has the same number of vertices and edges as the graph in Figure
1.15 a, but G is not isomorphic to these graphs.
Figure 1.16
G. SubGraphs
