Download Regular graphs - Shadows Government

MA2005 – Graphs and Networks
Graphs 1
1.1 Basic ideas
Aims:
To introduce some basic definitions and concepts of graph theory.
To present a useful result on the degree sequence of a graph.
A graph G consists of a finite non-empty set V(G) of elements called vertices together
with a finite set E(G) of unordered pairs of (not necessarily distinct) vertices called
edges.
Examples
(i) V(G) = {v1 , v2 , v3 , v4 , v5 }
E(G) = {v1v2 , v1v3 , v2 v3 , v2 v4 , v3 v4 , v3 v5 }
v1
v4
v2
v3
(ii) V(G) = {v1 , v2 , v3 , v4 , v5 }
v1
E(G) = {v1v2 , v2 v2 , v2 v3 , v2 v4 , v3 v4 (twice ) , v4 v5 }
v5
v4
v5
v2
v1
v3
v4
v6
(iii) V(G) = {v1 , v2 , v3 , v4 , v5 , v6 , v7 }
 v7
E(G) = {v1v2 , v2 v3 , v2 v4 , v3 v4 , v5 v6 }
v7 is an isolated vertex.
v2
v3
v5
If G has no loops (such as v2v2 in example (ii) or multiple edges (such as v3v4 in (ii)),
then G is called a simple graph. Thus the graphs in examples (i) and (iii) are simple,
that in (ii) is not, it is a non-simple graph.
Graphs (such as those in examples (i) and (ii) ) which ‘come in one piece’ are said to
be connected. The graph in (iii) is not connected: it is the union of three connected
subgraphs, called the components of the graph.
Two vertices u and v of a graph G are adjacent if there is an edge uv joining
them and we then say that u and v are incident with the edge (or that the edge is
incident with u and v). Similarly, two edges are adjacent if they have a vertex in
common.
Thus, for example, v1 and v2 are adjacent in the following graph, each being incident
with edge e1. We call v1 and v2 the end-vertices of e1. Also edges e1 and e2 are
adjacent.
v1
v4
e1
v2
e2
e3
e4
v3
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MA2005 – Graphs and Networks
Graphs 1
The degree of a vertex v, denoted by deg(v), is the number of edges incident with v.
Thus, in the graph above,
deg(v1) = 2 , deg (v2) = 2 , deg (v3) = 3 , deg (v4) = 1 .
In example (ii), deg (v2) = 5 (count the loop twice), deg (v3) = 3.
In example (iii), deg(v7) = 0. (An isolated vertex has degree 0.)
A vertex of degree 1 is called a leaf.
By the degree sequence of a graph, we mean the vertex degrees written in ascending
order with repeats where necessary. Thus the degree sequence in example (i) is
1,2,2,3,4, that in (iii) is 0,1,1,1,2,2,3.
Degree Sum Theorem. For a graph with vertex set {v1 , v2 ,, vn } and m edges,
n
 deg( v
k 1
k
)  2m .
Proof. Each edge contributes 2 to the sum of vertex degrees.
Note. As a consequence, in any graph the number of vertices of odd degree must be
even.
Isomorphism
Two graphs G1 and G2 are isomorphic if there is a one-one correspondence between
their vertex sets V(G1) and V(G2) such that for every pair of distinct vertices in G1
the number of edges joining them is equal to the number of edges joining their
corresponding vertices in G2.
Examples. The following two graphs are isomorphic with the following one-one
correspondence: v1  u1 , v2  u 4 , v3  u 2 , v4  u3 .
v3
u1
v1
v4
u4
v2
u2
u3
Likewise the following graphs are isomorphic:
v1
v2
v3
u4
u5
u2
v4
v5
v6
u3
u1
2
u6
MA2005 – Graphs and Networks
Graphs 1
1.2 Matrices associated with a graph
Consider a loopless graph G with vertex set {v1 , v2 ,, vn } and edge set
{e1 , e2 ,, em } . The adjacency matrix for G is the n  n matrix whose ijth element is
the number of edges joining vi to v j . The incidence matrix is the n  m matrix whose
ijth element is 1 if vi is incident with e j and 0 otherwise.
Example. For the graph
v1
e1
e5
v4
e6
e3
e4
e2
v2
v3
the adjacency matrix is
v1 v 2 v3 v 4
v1  0

v2  1
v3  0

v 4  1
1 0 1

0 1 1
1 0 2

1 2 0 
and the incidence matrix is
e1 e2 e3 e4 e5 e6
v1  1

v2  1
v3  0

v 4  0
0 0 0 1 0

1 0 0 0 1 .
1 1 1 0 0

0 1 1 1 1 
Exercises 1.1
1.
Draw an example of each of the following, each with 5 vertices and 8 edges:
(i)
(ii)
(iii)
2.
a simple graph,
a non-simple graph with no loops,
a non-simple graph with no multiple edges.
Which of the following are possible degree sequences of simple graphs. Draw
graphs for those that are.
(i) 1,2,2,3
(iv)1,2,3,3,4
(ii) 1,2,3,4
(v) 2,2,3,3,4,4
3
(iii) 0,0,1,1,2,2
(vi) 2,3,3,4,5,5.
MA2005 – Graphs and Networks
3.
Graphs 1
(a) Show that the following two graphs are isomorphic:
u4
u3
v1
u5
u2
v2
u1
v5
v3
v4
(b) Give a reason why the following two graphs cannot be isomorphic:
v1
v4
v5
v6
u4
u5
v7
u6
v2
4.
u1
v8
v3
u2
u8
u7
u3
(a) Write down the adjacency and incidence matrices for the following graph:
v1
v4
e1
e2
e3
e4
e5
v2
v3
e6
(b) Draw the graph with the adjacency matrix
0

1
1

1
1

1 1 1 1

0 1 0 1
1 0 0 0 .

0 0 0 1
1 0 1 0 
(c) Draw the graph with the incidence matrix
0

1
1

0
0

0 0 1 0 1 0

0 1 1 0 0 1
1 0 0 0 0 0 .

1 1 0 1 1 0
0 0 0 1 0 1 
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MA2005 – Graphs and Networks
Graphs 1
1.3 Some particular graphs
Null graphs
These are graphs with no edges and are denoted by Nn.
e.g. the null graph on 4 vertices




N4
Complete graphs
The complete graph on n vertices, denoted by Kn, is the simple graph in which every
pair of vertices is joined by an edge.
Examples:
K3
K4
K5
K6
Regular graphs
These are graphs in which every vertex has the same degree. For example, Kn is
regular of degree n-1. Regular graphs of degree 3 are called cubic graphs. The number
of edges of a regular graph with n vertices is given by n(n - 1)/2. The following
regular graph, called the Petersen graph is an interesting example.
Bipartite graphs
A graph G is bipartite if every vertex can be labelled with either a or b, so that every
edge is an ab edge. In these graphs, the vertex set is the union of two non-empty
disjoint sets A and B with each edge of the graph joining a vertex in A to a vertex in
B.
a
b
b
b
a
a
b
a
A
a
a

B
5
b
b
b
b
MA2005 – Graphs and Networks
Graphs 1
Complete bipartite graphs
Kr,s denotes the simple bipartite graph in which the sets A and B (as above) contain r
and s vertices and every vertex in A is adjacent to every vertex in B.
K3,4
or
The Platonic graphs
These are formed from the vertices and edges of the 5 regular (Platonic) solids:
Tetrahedron
Octahedron
Cube
Icosahedron
Dodecahedron
(From R. J. Wilson, Introduction to Graph Theory, Longman, 1996.)
The complement of a simple graph
The complement of a simple graph G is the simple graph with the same vertex set as G
with two vertices adjacent if and only if they are not adjacent in G. It is denoted by
G.
e.g.
if G is
then is G

1.4 More on degree sequences
Recall that, if a sequence d1 , d 2 ,, d n with d1  d 2    d n is the degree sequence
n
of a graph, then
d
j 1
j
must be even. Also, for a simple graph, d n  n  1 . However,
these two conditions are not sufficient for d1 , d 2 ,, d n to be the degree sequence of a
simple graph. For example, the sequence 1, 1, 3, 3 is not graphic.
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MA2005 – Graphs and Networks
Graphs 1
Useful result (Havel-Hakimi)
The sequence
d1 , d 2 ,, d n1 , j ,
where
d1  d 2    d n1  j  n  1 ,
is the degree sequence of a simple graph if and only if
d1 , d 2 ,, d n  j 1 , d n  j  1, d n  j 1  1, , d n 1  1
is (when rearranged in ascending order if necessary) the degree sequence of a simple
graph.
Example. From the sequence 2, 2, 3, 3, 4, 4 we derive 1, 2, 2, 2, 3 and 1, 1, 1, 1.
The last sequence is the degree sequence of
Working backwards, we obtain a graph for the original sequence: (non-isomorphic
answers are possible).

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MA2005 – Graphs and Networks
Graphs 1
Exercises 1.2
1.
Give (if it exists) an example of each of the following:
(i) A Platonic graph that is a complete graph.
(ii) A bipartite Platonic graph.
(iii) A bipartite graph that is regular of degree 4.
(iv) A cubic graph with 7 vertices.
2.
Determine the number of edges of the following graphs:
(i) K 8
(ii) K 5, 7
(iii) the Peterson graph.
3.
Draw the complements of the following graphs:
(i)
(ii)
4.
Draw, if possible, simple graphs with the following degree sequences:
(i) 2,3,3,4,5,5
(ii) 3,3,5,5,5,5
(iii) 2,2,3,3,4,5,5.
5.
Which of the following graphs are bipartite?
(i)
Petersen
(ii) C6 (6 – cycle):
(iii) C7 (7 – cycle):
(iv)
(v)
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