APPLICATIONS OF GRAPH THEORY IN HUMAN LIFE

INTERNATIONAL JOURNAL OF COMPUTER APPLICATION
ISSUE2, VOLUME 1 (FEBRUARY 2012) ISSN: 2250-1797
APPLICATIONS OF GRAPH THEORY
IN HUMAN LIFE
S. VENU MADHAVA SARMA
Assistant Professor of Mathematics
K. L. UNIVERSITY
Vaddeswaram
E-mail: svm190675@gmail.com
ABSTRACT
The author presents some graph theoretical planning techniques which have been employed
in the design of a GSM (Group Special Mobile) operated by the Bharat Sanchar Nigam
Limited. Apart from a new variant of the by now classical application of graph coloring to
frequency assignment, these techniques deal with the determination of the base station
identity code (BSIC), hopping sequence number (HSN), and location area code (LAC). It is
shown that GSM radio network planning involves a number of optimization problems and
Time Table Scheduling Problems which can be treated by graph theoretical methods.
Key Words :Bipartite Graph, Euler graph, Hamiltonian graph, Connected graph,
planner graph
1. Introduction:
The origin of graph theory started with the problem of Koinsber bridge, in 1735. This
problem lead to the concept of Eulerian Graph. Euler studied the problem of Koinsberg
bridge and constructed a structure to solve the problem called Eulerian graph. In 1840, A.F
Mobius gave the idea of complete graph and bipartite graph and Kuratowski proved that they
are planar by means of recreational problems. The concept of tree, (a connected graph
without cycles was implemented by Gustav Kirchhoff in 1845, and he employed graph
theoretical ideas in the calculation of currents in electrical networks or circuits. In 1852,
Thomas Gutherie found the famous four color problem. Then in 1856, Thomas. P. Kirkman
and William R.Hamilton studied cycles on polyhydra and invented the concept called
Hamiltonian graph by studying trips that visited certain sites exactly once. In 1913,
H.Dudeney mentioned a puzzle problem. Eventhough the four color problem was invented it
was solved only after a century by Kenneth Appel and Wolfgang Haken. This time is
considered as the birth of Graph Theory. Caley studied particular analytical forms from
differential calculus to study the trees. This had many implications in theoretical chemistry.
This lead to the invention of enumerative graph theory. Any how the term “Graph” was
introduced by Sylvester in 1878 where he drew an analogy between “Quantic invariants” and
covariants of algebra and molecular diagrams. In 1941, Ramsey worked on colorations which
lead to the identification of another branch of graph theory called extremel graph theory. In
1969, the four color problem was solved using computers by Heinrich. The study of
asymptotic graph connectivity gave rise to random graph theory.

1.1 Definition: A graph – usually denoted G(V,E) or G = (V,E) – consists of set of vertices
V together with a set of edges E. The number of vertices in a graph is usually denoted n while
the number of edges is usually denoted m.
1.2 Definition: Vertices are also known as nodes, points and (in social networks) as actors,
agents or players.
1.3 Definition: Edges are also known as lines and (in social networks) as ties or links. An
edge e = (u,v) is defined by the unordered pair of vertices that serve as its end points.
1.4 Example: The graph depicted in Figure 1 has vertex set V={a,b,c,d,e.f} and edge set E =
{(a,b),(b,c),(c,d),(c,e),(d,e),(e,f)}.
a b
c
d
e f
a b
c
d
e f
a b
c
d
e f
Figure 1.
1. 5 Definition: Two vertices u and v are adjacent if there exists an edge (u,v) that connects
them.
1.6 Definition: An edge (u,v) is said to be incident upon nodes u and v.
1.7 Definition: An edge e = (u,u) that links a vertex to itself is known as a self-loop or
reflexive tie.
1.8 Definition: Every graph has associated with it an adjacency matrix, which is a binary n n
matrix A in which aij = 1 and aji = 1 if vertex vi is adjacent to vertex vj, and aij = 0 and aji = 0
otherwise. The natural graphical representation of an adjacency matrix is a table, such as
shown below.
a b c d e f
a 0 1 0 0 0 0
b 1 0 1 0 0 0
c 0 1 0 1 1 0
d 0 0 1 0 1 0
e 0 0 1 1 0 1
f 0 0 0 0 1 0
Adjacency matrix for graph in Figure 1.

1.9 Definition: Examining either Figure 1 or given adjacency Matrix, we can see that not
every vertex is adjacent to every other. A graph in which all vertices are adjacent to all others
is said to be complete.
1.10 Definition : A subgraph of a graph G is a graph whose points and lines are contained in
G. A complete subgraph of G is a section of G that is complete
1.11 Definition : While not every vertex in the graph in Figure 1 is adjacent, one can
construct a sequence of adjacent vertices from any vertex to any other. Graphs with this
property are called connected.
1.12 Note: Reachability. Similarly, any pair of vertices in which one vertex can reach the
other via a sequence of adjacent vertices is called reachable. If we determine reachability for
every pair of vertices, we can construct a reachability matrix R such as depicted in Figure 2.
The matrix R can be thought of as the result of applying transitive closure to the adjacency
matrix A.
a b
c
d
e f
a b
c
d
e f
a b
c
d
e f
a b
c
d
e f
a b
c
d
e f
a b
c
d
e f
a b
c
d
e f
a b
c
d
e f
a b
c
d
e f
g
a b
c
d
e f
a b
c
d
e f
a b
c
d
e f
a b
c
d
e f
a b
c
d
e f
a b
c
d
e f
a b
c
d
e f
a b
c
d
e f
a b
c
d
e f
g
a b
c
d
e f
a b
c
d
e f
a b
c
d
e f
a b
c
d
e f
a b
c
d
e f
a b
c
d
e f
a b
c
d
e f
a b
c
d
e f
a b
c
d
e f
g
Figure: 2
1.13 Definition : A component of a graph is defined as a maximal subgraph in which a path
exists from every node to every other (i.e., they are mutually reachable). The size of a
component is defined as the number of nodes it contains. A connected graph has only one
component.
1.14 Definition : A sequence of adjacent vertices v0,v1,…,vn is known as a walk. In Figure 3,
the sequence a,b,c,b,c,g is a walk. A walk can also be seen as a sequence of incident edges,
where two edges are said to be incident if they share exactly one vertex.
1.15 Definition : A walk is closed if vo = vn.
1.16 Definition : A walk in which no vertex occurs more than once is known as a path. In
Figure 3, the sequence a,b,c,d,e,f is a path.
1.17 Definition : A walk in which no edge occurs more than once is known as a trail. In
Figure 3, the sequence a,b,c,e,d,c,g is a trail but not a path. Every path is a trail, and every
trail is a walk.

1.18 Definition : A cycle can be defined as a closed path in which n >= 3. The sequence c,e,d
in Figure 3 is a cycle.
1.19 Definition : A tree is a connected graph that contains no cycles. In a tree, every pair of
points is connected by a unique path. That is, there is only one way to get from A to B.
Figure 3: A labeled tree with
6 vertices and 5 edges
1.20 Definition : A spanning tree for a graph G is a sub-graph of G which is a tree that
includes every vertex of G.
1.21 Definition : The length of a walk (and therefore a path or trail) is defined as the number
of edges it contains. For example, in Figure 3, the path a,b,c,d,e has length 4.
1.22 Definition : The number of vertices adjacent to a given vertex is called the degree of the
vertex and is denoted d(v).
1.23 Definition : In the mathematical field of graph theory, a bipartite graph (or bigraph) is
a graph whose vertices can be divided into two disjoint sets U and V such that every edge
connects a vertex in U to one in V; that is, U and V are independent sets. Equivalently, a
bipartite graph is a graph that does not contain any odd-length cycles.
Figure 4: Example of a bipartite graph.

2. Euler Path and Example, Hamiltonian Path and Hamiltonian Circuit
2.1Definition : An Eulerian circuit in a graph G is circuit which includes every vertex and
every edge of G. It may pass through a vertex more than once, but because it is a circuit it
traverse each edge exactly once. A graph which has an Eulerian circuit is called an Eulerian
graph. An Eulerian path in a graph G is a walk which passes through every vertex of G and
which traverses each edge of G exactly once
2.2Example : Königsberg bridge problem: The city of Königsberg (now Kaliningrad) had
seven bridges on the Pregel River. People were wondering whether it would be possible to
take a walk through the city passing exactly once on each bridge. Euler built the
representative graph, observed that it had vertices of odd degree, and proved that this made
such a walk impossible. Does there exist a walk crossing each of the seven bridges of Königsberg
exactly once?
Figure 5: K¨onigsberg problem
2.3 Definition : Another closely related problem is finding a Hamilton path in the graph
(named after an Irish mathematician, Sir William Rowan Hamilton). Whereas an Euler path
is a path that visits every edge exactly once, a Hamilton path is a path that visits every vertex
in the graph 4 exactly once.
A Hamilton circuit is a path that visits every vertex in the graph exactly once and return to
the starting vertex. Determining whether such paths or circuits exist is an NP-complete
problem. In the diagram below, an example Hamilton Circuit would be
2.4 Example :
Figure 6: Hamilton Circuit would be AEFGCDBA.
3. Applications of GSM and Time Table Scheduling :
Graph theoretical concepts are widely used to study and model various applications, in
different areas. They include, study of molecules, construction of bonds in chemistry and the
study of atoms. Similarly, graph theory is used in sociology for example to measure actors

prestige or to explore diffusion mechanisms. Graph theory is used in biology and
conservation efforts where a vertex represents regions where certain species exist and the
edges represent migration path or movement between the regions. This information is
important when looking at breeding patterns or tracking the spread of disease, parasites and
to study the impact of migration that affect other species. Graph theoretical concepts are
widely used in Operations Research. For example, the traveling salesman problem, the
shortest spanning tree in a weighted graph, obtaining an optimal match of jobs and men and
locating the shortest path between two vertices in a graph. It is also used in modeling
transport networks, activity networks and theory of games. The network activity is used to
solve large number of combinatorial problems. The most popular and successful applications
of networks in OR is the planning and scheduling of large complicated projects. The best well
known problems are PERT(Project Evaluation Review Technique) and CPM (Critical Path
Method). Next, Game theory is applied to the problems in engineering, economics and war
science to find optimal way to perform
certain tasks in competitive environments. To represent the method of finite game a digraph
is used. Here, the vertices represent the positions and the edges represent the moves.
3.1 Traveling Salesman Problem :
TSP is a very well-known problem which is based on Hamilton cycle. The problem statement
is: Given a number of cities and the cost of traveling from any city to any other city, find the
cheapest round-trip route that visits every city exactly once and return to the starting city.
In graph terminology, where the vertices of the graph represent cities and the edges represent
the cost of traveling between the connected cities (adjacent vertices), traveling salesman
problem is just about trying to find the Hamilton cycle with the minimum weight. This
problem has been shown to be NP-Hard. Even though the problem is computationally
difficult, a large number of heuristics and exact methods are known, so that some instances
with tens of thousands of cities have been solved. The most direct solution would be to try all
permutations and see which one is cheapest (using brute force search). The running time for
this approach is O(V !), the factorial of the number of cities, so this solution becomes
impractical even for only 20 cities. A dynamic programming solution solves the problem with
a runtime complexity of O(V22V) by considering
dp[end][state] which means the minimum cost to travel from start vertex to end vertex using
the vertrices stated in the state (start vertex can be any vertex chosen at the start). As there are

V 2V subproblems and the time complexity to solve each sub-problems is O(V ), the overall
runtime complexity O(V22V).
3.2 Vertex Coloring:
Vertex coloring is one of the most important concepts in graph theory and is used in many
real time applications in computer science. Various coloring methods are available and can be
used on requirement basis. The proper coloring of a graph is the coloring of the vertices and
edges with minimal number of colors such that no two vertices should have the same color.
The minimum number of colors is called as the chromatic number and the graph is called
properly colored graph.
3.3 Map coloring and GSM mobile phone networks:
Groups Special Mobile (GSM) is a mobile phone network where the geographical area of this
network is divided into hexagonal regions or cells. Each cell has a communication tower
which connects with mobile phones within the cell. All mobile phones connect to the GSM
network by searching for cells in the neighbours. Since GSM operate only in four different
frequency ranges, it is clear by the concept of graph theory that only four colors can be used
to color the cellular regions. These four different colors are used for proper coloring of the
regions. Therefore, the vertex coloring algorithm may be used to assign at most four different
frequencies for any GSM mobile phone network. The authors have given the concept as
follows:
Given a map drawn on the plane or on the surface of a sphere, the four color theorem assets
that it is always possible to color the regions of a map properly using at most four distinct
colors such that no two adjacent regions are assigned the same color. Now, a dual graph is
constructed by putting a vertex inside each region of the map and connect two distinct
vertices by an edge iff their respective regions share a whole segment of their boundaries in

common. Then proper coloring of the dual graph gives proper coloring of the original map.
Since, coloring the regions of a planar graph G is equivalent to coloring the vertices of its
dual graph and vice versa. By coloring the map regions using four color theorem, the four
frequencies can be assigned to the regions accordingly.
3.4Time table scheduling:
Allocation of classes and subjects to the Teachers is one of the major issues if the constraints
are complex. Graph theory plays an important role in this problem. For „t‟ Teachers with „n‟
subjects the available number of „p‟ periods timetable has to be prepared. This is done as
follows. A bipartite graph (or bigraph is a graph whose vertices can be divided into two
disjoint sets U and V such that every edge connects a vertex in U to one in V; that is, U and V
are independent sets) G where the vertices are the number of Faculty say t1, t2, t3, t4, …….
tk and n number of subjects say n1, n2, n3, n4, ……. nm such that the vertices are
connected by „pi‟ edges. It is presumed that at any one period each Teacher can teach at most
one subject and that each subject can be taught by maximum one Teacher. Consider the first
period. The timetable for this single period corresponds to a matching in the graph and
conversely, each matching corresponds to a possible assignment of Teacher to subjects taught
during that period. So, the solution for the timetabling problem will be obtained by
partitioning the edges of graph G into minimum number of matching. Also the edges have to
be colored with minimum number of colors. This problem can also be solved by vertex
coloring algorithm. “ The line graph L(G) of G has equal number of vertices and edges of G
and two vertices in L(G) are connected by an edge iff the corresponding edges of G have a
vertex in common. The line graph L(G) is a simple graph and a proper vertex coloring of
L(G) gives a proper edge coloring of G by the same number of colors. So, the problem can be
solved by
finding minimum proper vertex coloring of L(G).” For example, Consider there are 4
Teachers namely t1, t2, t3, t4,. and 5 subjects say n1, n2, n3, n4, n5 to be taught. The
teaching requirement matrix p = [pij] is given as.

P n1 n2 n3 n4 n5
t1 2 0 1 1 0
t2 0 1 0 1 0
t3 0 1 1 1 0
t4 0 0 0 1 1
Figure – 7: The teaching requirement matrix for four Teachers and five subjects
The bipartite graph is constructed as follows.
Figure – 8: Bipartite graph with 4 Teachers and 5 subjects
Finally, the authors found that proper coloring of the above mentioned graph can be done by
4 colors using the vertex coloring algorithm which leads to the edge coloring of the bipartite
multigraph G. Four colors are interpreted to four periods
…. 1 2 3 4
t1 n1 n2 n3 n4
Figure – 9: The schedule for the four subjects
References:
[1] Gian Luca Marcialis, Fabio Roli, Alessandra Serrau, “Graph Based and Structural
Methods for Fingerprint Classification, Springer verlag, Berlin Heidelberg 2007
[2] John.P.Hayes, “A graph Model for Fault Tolerant Computing Systems”, IEEE September
1976
[3] Narasingh Deo, “Graph theory with applications to engineering and computer science”,
Prentice Hall of India, 1990.

[4] Perri Mehonen, Janne Riihijarvi, Marina Petrova, “Automatic Channel allocation for
small wireless area networks using graph coloring algorithm approach”, IEEE 2004
[5] Shariefuddin Pirzada and Ashay Dharwadker, “Journal of the Korean Society for
Industrial and applied Mathematics, Volume 11, No.4, 2007
[6] Sven Dickinson, Pelillo, Ramin Zabih, “Introduction to the special section on graph
algorithms in computer vision”, IEEE on pattern analysis, Vol 23 No. 10, September 2001
[7] V.P.Eswaramoorthy, “New algorithm for analyzing performance of neighbourhood
strategies in solving job shop scheduling problems,
[8] International Journal of Engineering Science and Technology Vol. 2(9), 2010, 4610-4621.
[9] International Journal of Mathematical Archieve-2(11),2011,Page 2113-2118.

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