Chapter 1 Introduction to Graph Theory

Ali Mahmudi, Introduction to Graph Theory [1]

Chapter 1 Introduction to Graph Theory

Dr. Ali Mahmudi (Jurusan Pendidikan Matematika FMIPA UNY)

To those who ask what the infinitely small quantity in mathematics is,

we answer that it is actually zero. Hence there are not so many mysteries hidden in this concept

as they are usually believed to be.

(Leonhard Euler (1707-1783)

A. The Konigsberg Problem

Konigsberg is a city which was the capital of East Prussia but now is

known as Kaliningrad in Russia. The city is built around the River Pregel where

it joins another river. An island named Kniephof is in the middle of where the

two rivers join. There are seven bridges that join the different parts of the city on

both sides of the rivers and the island as shown in the Figure 1.1.

(Sumber: http://math.youngzones.org/Konigsberg.html)

Figure 1.1. Konigsberg Bridge


People tried to find a way to walk all seven

bridges without crossing a bridge twice. This

problem was called Konigsberg bridge problem: is

there a walking route that crosses each of the

seven bridges of Konigsberg exactly once? The

problem came to the attention of a Swiss

mathematician named Leonhard Euler

(pronounced "oiler").

In 1735, Euler presented the solution to the problem. He explained why

crossing all seven bridges without crossing a bridge twice was impossible. While

solving this problem, he developed a new mathematics field called graph theory,

which later served as the basis for another mathematical field called topology.

The key to Euler’s solution was very simple abstraction of the problem. Euler

redraw the diagram of the city of Konigsberg by representing each of the land

masses as a vertex and representing each bridge as an edge connecting the vertices

corresponding to the land masses as shown in the Figure 1.2. By this method we

have a graph that encodes the necessary information. The problem reduces to

finding a “closed walk” in the graph which traverses each edge exactly once. This

is called an Eulerian circuit.

Figure 1.2. Graph as a Model of Konigsberg Problem

Leonhard Euler (1707-1783) (Sumber: http://www.pdmi.ras.ru/)


The field of graph theory began to blossom in the twentieth century as more

and more modeling possibilities we recognized – and growth continues. In the

mid 1800s, people began to realize that graphs could be used to model many

things that were of interest in society. For example, the “Four Color Maps

Conjecture”, introduced by DeMorgan in 1852, was a famous problem that was

seemingly unrelated to graph theory. The conjecture stated that four is the

maximum number of colors required to color any map where bordering regions

are colored differently. This conjecture can easily be phrased in terms of graph

theory, and many researchers used this approach during the dozen decades that the

problem remained unsolved.

B. The Definition of Graph

In a football league there are eight teams, which we denote by A, B, C, D, E,

F, G, and H. After a few weeks of the season the following games have been

played.

A has played F and H B has played E, F, and H C has played G and H D has played E and G

E has played B, D, and G F has played A and B G has played C and E H has played A, B, and C

We may illustrate this situation by either of the two diagram of Figure 1.3,

where the teams are represented by dots and two such dots are joined by a line

whenever the corresponding teams have played each other.

Figure 1.3. Games in the Football League

A B

C

D

E F

G

H

A B

C

D

E F

G

H


In the diagram on the left, the dots have been joined using straight lines,

while in the other diagram three of lines used are not straight. But since we are

only interested in which games have been played, it does not matter whether the

lines are straight or not.

The diagrams may be used to describe other situation. For example, the

eight dots represent eight people, with le line joining a pair of dots if the two

people know each other. Or, the dots could be communication centre with the

lines denoting communication links. Indeed, as we will see later, many real-world

situations can conveniently be described by means of such drawing, which we call

graphs. A graph can be thought of as a drawing or diagram consisting of

collection of vertices (dots or points) together with edges (lines) joining certain

pairs of these vertices, as in the diagrams of Figure 1.2. Here is formal definition

of graph.

If u and v are two vertices in G and e = (u,v) is edge in G, then

(i) vertex u and v are endpoints of edge e

(ii) vertex u and v are adjacent

(iii) vertex u and v incident to e or e incident to vertex u and v

Consider that both graph of Figure 1.3 represent an equal graph. The vertex

set of the graph is V = {A, B, C, D, E, F, G, H} and the edge set is E = {(A,F),

(A,H), (B,E), (B,F), (B,H), (C,G), (C,H), (D,E), (D,G), (E,G)}. We can also notate

the edge set as E = {AF, AH, BE, BF, BH, CG, CH, DE, DG, EG}.

Definition 1.1

A graph G = ((V(G), E(G)) consist two sets, V(G) and V(G). V(G) is the vertex

set of the graph, often denoted by just V, which is a nonempty set of elements

called vertices. E(G) is the edge set of the graph, often denoted by just E, which

is a possibly empty set of elements called edges, such that each edge e in E is

assigned an unordered pair of vertices (u, v), called the end vertices of e.


Example 1.1

Let G = (V(G), E(G)) is a graph, where V(G) = {1, 2, 3, 4} and E(G) = {(1,2),

(1,3), (1,4), (2,4)}. We can represent the graph as follow.

Figure 1.4. A Graph with Four Vertices and Five Edges

C. Graph as a Model

Graph theory provides a natural mathematical model for some problems. At

this chapter we only describe the problem and delay detailed discussion of their

solution to later chapters.

Problem 1

Suppose that the graph of Figure 1.5 represents a network of telephone lines and

poles. We are interested in the networks vulnerability to accidental disruption. We

want to identify those lines and poles that must stay in service to avoid

disconnecting the network.

Figure 1.5. Network of Telephone Line

2 1

4 3

D A

B

C

E

F

e1

e2

e3 e4

e5

e6

e7

e8


There is no single line whose disruption (removal) will disconnect the graph

(network), but the graph will become disconnected if we remove the two lines

represented by edges e4 and e5, for example. When it comes to poles, the network

is more vulnerable since there is a single vertex, vertex d, whose removal

disconnects the graph. This illustrates the notions of edge connectivity and vertex

connectivity of a graph.We may also want to find a smallest possible set of edges

needed to connect the six vertices. There are several examples of such minimal

sets. One is {e1, e3, e5, e6, e7}

Problem 2

Suppose we have three houses each of which have to be supplied with electricity,

gas and water. Is it possible to connect each utility with each of the three houses

without the lines or mains crossing?

We can represent the connection of the three houses to the three utilities by

the graph of Figure 1.6. Here we have a graph with six vertices, three of which

represent the houses (denoted by H1, H2, H3), the other three represent the utilities

(denoted by E, G, W), and an edge joins two vertices if and only if one vertex

denotes a house and the other vertex a utility.

Figure 1.6 The Three Utilities Graph

The problem then is as the whether or not we can draw this graph in such a

way that no two edges intersect. The answer is no – we will see why when we

look at planar graphs in Chapter 4.

E

H1 H2 H3

G W


Problem 3

A company has branches in each of six cities C1, C2, C3, …, C6. The airfare for a

direct flight from Ci to Cj is given by the (i, j)th entry of the following matrix

(where indicates that there is no direct flight). For example the fare from C1 to

C4 is IDR 400.000,00 from C2 to C3 is IDR 1.500.000,00.

05525251055010202525100102040

2010015252015050102540500

The company is interested in computing a table of cheapest fares between

pairs of cities. (Even if there is a direct flight between two cities this may not be

the cheapest route). We can first represent the situation by a weighted graph, i.e., a

graph with “weights” attached to edges according to the airfares, as in Figure 1.7.

Figure 1.7 The Weighted Graph of Airfare for Direct Flights Between Six Cities

The problem can then be solved using Dijkstra’s algorithm. This type of problem is called a shortest path problem.

50

A

B

C

D

E

F

15

10 10

55

10

25

25 20 20

25

40


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 N(v). Given graphs G = (V(G), E(G)), where V(G) = {v1, v2, v3, v4} and

E(G1) = {v1v2, v1v3, v3v4}. 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, e2 and e3 are multiple-edges, edge e5 is loop, and

E is an isolated vertex. The neighborhood of vertex B, D, and E, are N(B) = {A,

C}, N(D) = {C}, and N(E) = { } respectively.

1. Simple Graph and Multi-Graph

E

D C

B A

e2 e3

e1

e5 e4

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.


2. Complete Graph

If the complete graph has vertices v1, v2, …, vn then the edge set can be

given by

E = {(vi, vj); vi vj; i,j = 1, 2, …, n}

Note that the grah has 1nn21

edges, since there are n – 1 edges incident

with each of the n vertices vi, so a total of n(n – 1), but divide by 2, since (vi, vj) =

(vj, vi). A complete graph with n vertices is denoted by Kn. Figure 1.9 shows K1,

K2, …, K5.

Figure 1.9 The Complete Graphs on at most five vertices

3. Empty Graph

K1 K2

K4 K5

K3

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.


An empty graph with n vertices is denoted by Nn. Figure 1.10 shows N1, N2,

…, N5.

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

N1

N2

N3

N4

N5

Definition 1.5

Let G be a graph. If the vertex set V(G) can be partitioned into two

nonempty subsets X and Y, i.e. X Y = V(G) 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

V(G) = 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 Km,n.


Consider that since each of the m vertices in the partition set X of Km,n is

adjacent to each of the n vertices in the partition set Y, Km,n has m x n edges.

Figure 1.12 shows complete bipartite graphs K2,3 and K3,3.

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

K2,3 K3,3

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 d(v) or dG(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.


Minimum and maximum degree of G, denoted (G) dan (G) respectively, are defined below.

(G) = minimum {d(v) | v V(G)}

(G) = maxsimum {d(v) | v V(G)}

Consider Figure 1.13 below.

Figure 1.13

In the graph of Figure 1.13 we have d(a) = 1, d(b) = 3, d(c) = 4, d(d) = 3,

d(e) = 1, and d(f) = 0. We also have (G) = 0 and (G) = 4. Note that d(a) + d(b) +

d(c) + d(d) + d(e) + d(f) = 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(Vv

where |E(G)| = the number of edges in the graph.


Proof

Let A is the set of odd vertices of G and B is the set of even vertices of G. Then,

for each aA, d(a) is even and soAa

)a(d , being a sum of even numbers, is even.

By Theorem 1.2, we get

Aa

)a(d + Bb

)b(d = )G(Vv

)v(d = 2|E(G)|

Thus

Bb

)b(d = 2|E(G)| – Aa

)a(d is even (being the difference of two even numbers).

As all the terms in Bb

)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

GVGE

(G),

where |V(G)| = number of vertices of G.

Definition 1.5

If for some positive integer k, d(v) = k for every vertex v of the graph G, then G

is called k-regular.


A complete graph is one that is k-regular for some k. The graph drawn

below graph G is a 2-regular and graph K4 is a 3-regular.

Figure 1.14 Regular.Graph

Notice that the complete graph Kn is (n-1)-regular. The complete bipartite

graph Kn,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 G1 and G2 are isomorphic if the vertices of G1 can

be paired off with the vertices of G2 and the edges of G1 can be paired off with the

edges of G2 in such way that the ends paired off edges are paired off. Thus G1 is

really just the same graph as G2, apart from a possible change in how the vertices

2-regular 3-regular

Definition 1.6

A graph G1 = (V1, E1) is said to be isomorphic to the graph G2 = (V2, E2) if there is one-to one correspondence between the edge sets E1 and E2 in such a way that if e1 is an edge with end vertices u1 and v1 in G1 then the corresponding edge e2 in G2 has its end points the vertices u2 and v2 in G2 which correspond to u1 and v1 respectively. Such a pair of correspondence is called a graph isomorphism.


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

G1 and G2 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


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

Fo example, in Figure 1.17 H1 is subgraph of H and H2 is spanning

subgraph of G.

Figure 1.17 Graph and Subgraph

1H 2H

K

L N M

Definition 1.7

Let H be a graph with vertex set V(H) and an edge set E(H). Similarly let G be

a graph with vertex set V(G) and edge set E(G). H is called subgraph of G if

V(H) )G(V and ).G(E)H(E If V(H) = V(G) then H is spanning subgraph of G.

We also say that G is a supergraph of H.

G

Z X

V W W V

Z X

Y

Z

X

W V

Y


From the definition we can conclude that any simple graph on n vertices is

subgraph of the complete graph Kn. Notice that we can get a subgraph of any

graph by deleting of a vertex or an edge the graph.

H. Complement of Graph

In the Figure below, Gc is complement of G.

Figure 1.18 Graph and Complement

Consequently If G is a complete graph, then Gc is a empty or null graph and

for any graph, G Gc is a complete graph.

I. Representation of Graph

We can represent a simple graph G = (V(G),E(G)) in many different ways.

We have seen that we can represent it by describing V and E, or by drawing the

graph. Another possible representation is an adjacency matrix.

Definition 1.8

Let G a simple graph. Complement of G, denoted Gc, is a simple graph in

such that (V(G) = V(Gc)) and two vertices u is adjacent to v in Gc if and only

if u is not adjacent to v in G

A

E B

D C

A

B

C D

E

G Gc


1. Adjacency Matrix

Let G a graph with V(G) = nvvvv ...,,, 321 . Adjacency matrix of G is a

(symmetric) matrix A(G) = ija , where ija state the number of edges adjacent to

vertices iv and jv .

2. Incidency Matrix

Let G a graph with n vertices v1, v2, …, vn and k edges. Incidency matrix of

G is a matrix I(G) = ijm that has ordo n x k, where:

0, if edge ej is not incident to vertex vi

mij = 1, if edge ej incident to vertex vj and ej is not loop

2, if edge ej incident to vertex vi and ej is loop

Let G a graph represented by diagram below.

Figure 1.19

Adjacency matrix and incidency of graph G above are as follows

v1 v2 v3 v4 v1 1 1 2 0 A(G) = v2 1 1 1 1 v3 2 1 0 0 v4 0 1 0 0 e1 e2 e3 e4 e5 e6 e7 v1 2 1 1 1 0 0 0 v2 0 1 0 0 1 2 1

M(G) = v3 0 0 1 1 1 0 0 v4 0 0 0 0 0 0 1

G

e1 v1

e4 e2 e3

e7 e5

e6

v3 v2

v4


Based on the examples above we can notice the following characteristic.

1. Adjacency matrix is symmetric to main diagonal

2. If a graph has no loop, then each element of diagonal of adjacency matrix is

zero.

3. Element of matrix of simple graph is 0 or 1

4. Degree of vertex vi in G correspond to sum of all element of row (column) i of

adjacency matrix if element of main diagonal i is multiplied by 2.

5. Degree of vertex vi in G correspond to sum all element of row i of incidency

matirx

6. Sum all element of every column of incidency matrix is 2.

Exercise

1. Given graph G below.

a. Give a label to vertices of the graph b. Determine a vertex and a edge set of the graph c. Is this a simple graph?

2. Give an example of graph with 5 vertices, 5 edges, and 1 isolated vertex. Is this a simple graph?

3. Give an example of graph with 4 vertices, 6 edges, and 1 isolated vertex. Is this a simple graph?

4. A graph G has 17 vertices and.maximum vertex degree of G is 3. Determine the maximum number of edges of G.

5. A graph G has 23 edges and minimum vertex degree of G is 5. Determine minimum the number of vertices of G.

6. A graph G has 19 edges and minimum vertex degree of G is 3. Determine maximum the number of vertices of G.

7. A graph G has 17 edges and minimum vertex degree of G is 5. Determine the minimum number of vertices of G.


8. A graph G has 11 edges and minimum vertex degree of G is 5. Determine maximum of vertices of G.

9. Find a pair of isomorphic from three graphs below.

10 Identify whether the following pair of graph isomorphism

11. Let G be a graph in which there is no pair of adjacent edges. What can you

says about the degrees of the vertices in G?

12. Let G be a k-regular graph, where k is an odd number. Prove that the number

of edges in G is a multiple of k.

13. Let G be a graph with n vertices and exactly n – 1 edges. Prove that G has

either a vertex of degree 1 or an isolated vertex.

14. Prove that there is no simple graph with six vertices, one of which has degree 2

two have degree 3, three have degree 4 and the remaining vertex has degree 5.


15. Represent the figure below to a graph.

16. Suppose G = (V(G), E(G)) wehere V(G) = {A, B, C, D, E} and E(G) = {BD,

AE, CB}. Determine adjacency matrix and incideny matrix of G.

17. Suppose a graph has the adjaceny matrix shown below. Draw the graph and

formally described.

0110010010100110110100110

18. Suppose a graph has the incidency matrix shown below. Draw the graph and

formally described.

11000100110110100110


BIBLIOGRAPHY

Budayasa, K. (1995). Matematika Diskret I. Universitiy Press IKIP Surabaya

_________. (1996). Pengantar Teori Graf. Makalah disajikan Pada Kursus Pendalaman Materi SMU di PPPG Matematika Yogyakarta, Tanggal 27 Oktober 1996 – 6 Nopember 1996.

Balakrishnan, V.K. (1995). Combinatorics. USA: Schaum Outline Series. McGraw-Hill, INC.

Clark, J. and Holton, D.A. (1991). A First Look At Graf Theory. World Scientific Publishing Co., Singapore.

Grimaldi, R.P. (1999). Discrete and Combinatorial Mathematics an Applied Introduction. Fourth Edition. USA: Addision-Wesley.

Harris, J.M., Hirst, J.M., & Mossinghoff, M.J. (2008). Combinatorics and Graph Theory. Secon Edition. Springer. USA: Springer.

Lovasz, L., Pelikan, J., & Vesztergombi. (2000). Discrete Mathematics. Elementary and Beyond. USA: Springer.

Wilson, J.R. and Watkinsons, J.J. (1990). Graf (An Introductory Approach). Alih Bahasa Oleh Theresia MH Tirta Seputra. University Press IKIP Surabaya Tahun 1992

Chapter 1 Introduction to Graph Theory

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