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2. 2
Introduction
Graph theory, branch of mathematics concerned with
networks of points connected by lines. The subject of graph theory had its
beginnings in recreational problems, but it was grown into significant area of
mathematical research with application in chemistry, computer science, and social
science. Graphs can be used to model many types of relations and processes in
physical, biological, social and information systems. Many practical problems can
be represented by graphs.
Application of Graph Theory:
Graphs can be used to model many types of relations and processes in
physical, biological, social and information systems. Many practical problems can be
represented by graphs.
In computer science
In chemistry
In biology
In engineering
In sociology
In linguistics

3. 3
History
Bridges in Königsberg
.
A map of Königsberg showing the seven bridges
This story is about the Swiss mathematician Leonhard Euler (1707 –
1783). While he lived in the Prussian town of Königsberg at the Baltic Sea, he was
thinking about the following problem. Königsberg is divided into four parts by the
river Pregel, and connected by seven bridges. Is it possible to tour Königsberg along
a path that crosses every bridge once, and at most once? You can start and finish
wherever you want, not necessarily in the same place.
Trying various paths very quickly suggests that it is impossible: you
always end up with a bridge you haven’t crossed, but with no way to get there
without crossing another bridge twice. But it would take a very long time to
try every possible path.
Having solved the Drawing Graph problem, we can now solve the
Königsberg Bridges problem by converting map of Königsberg into graph. Every
island is represented by a vertex, and every bridge connecting two islands is
represented by an edge connecting the two points.Here is what we get,

4. 4

5. 5
Application of Graph Theory in Computer Science:
The use of mathematics is quite visible in every area of
computer science i.e. in artificial intelligence, software development environments
and tools, software architecture and design, multiprocessing, automatic control,
distributed and concurrent algorithms etc. Mathematics helps in the design,
implementation and analysis of algorithms for scientific and engineering
applications. It also improves the effectiveness and applicability of existing methods
and algorithms.
(1) Shortest Path in Road or a Network :
A road network can be considered as a graph with positive weights.
The nodes represent road junctions and each edge of the graph is associated with a
road segment between two junctions. The weight of an edge may correspond to the
length of the associated road segment, the time needed to traverse the segment or the
cost of traversing the segment. Using directed edges it is also possible to model one-
way streets. Such graphs are special in the sense that some edges are more important
than others for long distance travel (e.g. highways). This property has been
formalized using the notion of highway dimension. There are a great number of
algorithms that exploit this property and are therefore able to compute the shortest
path a lot quicker than would be possible on general graphs.
All of these algorithms work in two phases. In the first phase, the graph
is preprocessed without knowing the source or target node. This phase may take
several days for realistic data and some techniques. The second phase is the query
phase. In this phase, source and target node are known. The running time of the
second phase is generally less than a second. The idea is that the road network is
static, so the preprocessing phase can be done once and used for a large number of
queries on the same road network.
Now this system installed in of vehicles which are
connected with internet and u can easily locate the destination and also find
the shortest path. This method is also applicable in networking e.g. social
networking site like facebook, twitter and etc. This technique is also used
when we connected with online libraries. Figures are on next page

6. 6

7. 7
(2) Computer Network Security:
A team of computer scientists led by Eric Filiol at the Virology and
Cryptology Lab, ESAT, and the French Navy, ESCANSIC, have recently used the
vertex cover algorithm to simulate the propagation of stealth worms on large
computer networks and design optimal strategies for protecting the network against
such virus attacks in real-time.
Figure 2.1. The set {2, 4, 5} is a minimum vertex cover in this computer network
network and showed that that the combinatorial topology of routing may have a
huge impact on the worm propagation and thus some servers play a more essential
and significant role than others. The real-time capability to identify them is
essential to greatly hinder worm propagation. The idea is to find a minimum vertex
cover in the graph whose vertices are the routing servers and whose edges are the
(possibly dynamic) connections between routing servers. This is an optimal
solution for worm propagation and an optimal solution for designing the network
defense strategy. Figure 2.1 above shows a simple computer network and a
corresponding minimum vertex cover {2, 4, 5}.

8. 8
(3) Map Coloring and GSM Mobile Phone Networks:
Given a map drawn on the plane or the surface of a sphere, the
famous four color theorem asserts that it is always possible to properly color the
regions of the map such that no two adjacent regions are assigned the same color,
using at most four distinct colors. For any given map, we can construct its dual graph
as follows. Put a vertex inside each region of the map and connect two distinct
vertices by an edge if and only if their respective regions share a whole segment of
their boundaries in common. Then, a proper vertex coloring of the dual graph yields
a proper coloring of the regions of the original map.
We use the vertex coloring algorithm to find a proper coloring of the
map of India with four colors, see figures 3.1 and 3.2 above.
The Grouped Special Mobile (GSM) was created in 1982 to provide
a standard for a mobile telephone system. The first GSM network was launched in
1991 by Radiolinja in Finland with joint technical infrastructure maintenance from
Ericsson. Today, GSM is the most popular standard for mobile phones in the world,
used by over 2 billion people across more than 212 countries. GSM is a cellular
Figure 3.1. The
map of India
Figure 3.2. The dual
graph of the map of
India

9. 9
network with its entire geographical range divided into hexagonal 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 four different
frequency ranges. The reason why only four different frequencies suffice is clear: the
map of the cellular regions can be properly colored by using only four different
colors! So, the vertex coloring algorithm may be used for assigning at most four
different frequencies for any GSM mobile phone network, see figure 3.3 below.
Figure 3.3. The cells of a GSM mobile phone network

10. 10
(4) Bi-processing task:
Assume that there is a set of processors and set of tasks. Each task
has to execute on two processors simultaneously and these two processors must be
pre assigned to the task. A processor cannot work on two jobs simultaneously. This
type of task will arise when scheduling of file transfers between processors or in
case of mutual diagnostic besting of processors. This can be modeled by considering
a graph whose vertices corresponding to the processes and if there is any task that
has to be executed on processors I and j, then and edge to be added between the two
vertices. Now the scheduling problem is to assign colors to edges in such a way that
every color appears at most once at a vertex.
If there are no multiple edges in graph i.e. no two task require the
same two processors then the edge coloring technique can be adopted. The algorithm
(given below) developed for multiple edges.
The diagram shows the tasks namely task1, task2, task3, and task4
allocated to processors (P1,P2), (P1,P6), (P2,P4) and (P3,P7) respectively.
There is a lot of other application of graph theory in computer science
some of them are mention above

11. 11
Application of graph theory in Chemistry:
Chemical graph theory is the topology branch of mathematical
chemistry which applies graph theory to mathematical modelling of chemical
phenomena. The pioneers of the chemical graph theory areAlexandru Balaban, Ante
Graovac, Ivan Gutman, Haruo Hosoya, Milan Randić and Nenad Trinajstić
(also Harry Wiener and others). In 1988, it was reported that several hundred
researchers worked in this area producing about 500 articles annually. A number of
monographs have been written in the area, including the two-volume comprehensive
text by Trinajstic, Chemical Graph Theory, that summarized the field up to mid-
1980s.
The adherents of the theory maintain that the properties of a chemical
graph i.e. a graph-theoretical representation of a molecule gives valuable insights
into the chemical phenomena. The opponents contend that graphs play only a fringe
role in chemical research. One variant of the theory is the representation of materials
as infinite Euclidean graphs, particularly crystals by periodic graphs.
Graph is also helpful in constructing the molecular structure as well as
lattice of the molecule. It also helps us to show the bond relation in between atoms
and molecules, also help in comparing structure of one molecule to other.

12. (1) Molecular Graph
In chemical gr
graph or chemical graph is a
compound in terms of graph
vertices correspond to the at
bonds. Its vertices are labele
are labeled with the types of
be ignored.
A hydrogen-d
molecular graph is the mole
Molecular gra
compounds which have the
such as iso-pentane and neo
normally does not contain a
of the bonds, and therefore c
as cis and trans 2-butene) or
In some impor
following classical definitio
graph one-to-one correspon
vertices of the graph corresp
correspond to chemical bon
materials as infinite Euclide
Cis 2-butene
12
aph:
l graph theory and in mathematical chemi
is a representation of the structural formu
aph theory. A chemical graph is a labeled
e atoms of the compound and edges corre
eled with the kinds of the corresponding
of bonds. For particular purposes any of
depleted molecular graph or hydrogen-s
olecular graph with hydrogen vertices del
graphs can distinguish between structural
he same molecular formula but non-isom
eo-pentane. On the other hand, the molec
n any information about the three-dimens
re cannot distinguish between conformati
) or stereoisomer (such as D- and L-glyce
portant cases (topological index calculatio
ition is sufficient: molecular graph is conn
onded to structural formula of chemical c
espond to atoms of the molecule and edge
onds between these atoms. One variant is
idean graphs, in particular, crystals as per
trans 2-buten
mistry, a molecular
mula of a chemical
led graph whose
rrespond to chemical
ng atoms and edges
of the labelings may
suppressed
deleted.
ral isomers,
morphic graphs -
lecular graph
nsional arrangement
ational isomers (such
ceraldehydes).
ation etc.) the
onnected undirected
l compound so that
dges of the graph
t is to represent
periodic graphs.
tene

13. (2) Periodic Graph
In crystallogra
dimensional periodic graph
or nodes are points in three
or spacers) are line segment
independent axial directions
vertices are uniformly discre
any two vertices. The vertic
clusters of atoms such as sin
building units, while each ed
Although the n
mathematical (actually a cry
covering graph over a finite
space (or honeycomb) in the
contemporary effort in the a
(design), including metal-or
neo-pentane
13
h (Crystallography):
graphy, a periodic graph or crystal net is
ph, i.e., a three-dimensional Euclidean gr
-dimensional Euclidean space, and who
ents connecting pairs of vertices, periodic
ons. There is usually an implicit assumpti
screte, i.e. that there is a fixed minimum d
tices may represent positions of atoms or
single-metal ions, molecular building blo
h edge represents a chemical bond or a po
e notion of a periodic graph or crystal ne
crystal net is nothing but a periodic realiz
ite graph ), and is closely related to that o
the theory of polytopes and similar areas
e area is motivated by crystal engineering
organic frameworks (MOFs) and zeolite
Iso-pen
is a three-
graph whose vertices
hose edges (or bonds
dic in three linearly
ption that the set of
distance between
or complexes or
blocks, orsecondary
polymeric ligand.
net is ultimately
alization of an abelian
t of a Tessellation of
eas, much of the
ing and prediction
ites.
entane

14. A (large
repres
A (large
balls are
14
rge) unit cell of Diamond crystal net; the balls
resent carbon atoms and the sticks represent
covalent bonds
rge) unit cell of Quartz A crystal net; the black
are silicon atoms and the red ones are oxygen.

15. 15
Application of Graph Theory in Biology:
Biological networks come in a variety of forms. Nodes in biological
networks represent bimolecular such as genes, proteins or metabolites, and edges
connecting these nodes indicate functional, physical or chemical interactions
between the corresponding bimolecular. Understanding these complex biological
systems has become an important problem that has lead to intensive research in
network analyses, modeling, and function and disease gene identification and
prediction. The hope is that utilizing such systems-level approaches to analyzing and
modeling complex biological systems will provide insights into the inner working of
the cell, biological function, and disease.
1. transcriptional regulation networks:
In transcriptional regulation networks, nodes represent genes and edges
are directed from a gene that encodes for a transcription factor protein to a gene
transcriptional regulated by that transcription factor (see Figure). Thus, the network
structure is an abstraction of the system’s biochemical dynamics that is responsible
for regulating the expression of genes in cells. The two best characterized
transcriptional regulation networks are those of a eukaryote, the yeast and a
bacterium.
Ss
Figure: two example of biological network.

16. 16
2. Metabolic networks:
One of the most important life processes is the metabolism of an
organism, the basic chemical system that generates essential components such as
amino acids, sugars and lipids, and the energy required to synthesize them and to use
them in creating proteins and cellular structures. A metabolic network represents this
system of connected chemical reactions, i.e., the complete set of metabolic and
physical processes that determine the physiological and biochemical properties of a
cell. Metabolism network reconstruction breaks down metabolism pathways into
their respective reactions and enzymes. Thus, in these networks, small-molecule
substrates can be envisioned as nodes and the links as the enzyme-catalyzed
reactions that transform one metabolite into another. With the sequencing of
complete genomes, it is now possible to reconstruct the network of biochemical
reactions in many organisms, from bacteria to human. These networks are available
in several databases, such as Kyoto Encyclopedia of Genes and Genomes (KEGG).
Metabolic networks are powerful tools for studying and modeling
metabolism. However, graph theoretic description of real-world metabolic networks
(Figure 2.1 A) still needs to be established precisely. For example, in the most
abstract approach, all interacting metabolites are considered equally and the edges
between nodes represent reactions that convert one substrate into another (Figure 2.1
B). However, for many biological applications, it is useful to ignore co-factors,
which can result in a completely different type of mapping that connects only the
main source metabolites to the main products (Figure 2.1 C).
Fig 2.1:Different graph theory representation of metabolic network
Fig 2.1A: real world metabolic path way
Fig 2.1 B: representation when all interacting metabolic are consider equally
Fig 2.1 C: representation when only the main source metabolites and the main
products are considered

17. 17
3. PPI Network
Finally, in protein-protein interaction (PPI) networks, nodes correspond
to proteins and undirected edges represent physical interactions amongst them. PPI
networks represent an opportunity as well as the challenge. Analyzing these
networks may provide useful clues about the function of individual proteins, protein
complexes, and larger cellular machines. However, PPI data volume and noisiness is
making many algorithms for its analyses intractable. Additionally, graph
representation of PPI data with nodes and edges corresponding to proteins and
protein interactions, respectively, does not address some of the major properties of
protein interaction data. It does not deal with the noisiness of the data, i.e., the large
number of false positives and negatives. Moreover, all spatial and temporal
information is lost, as well as the information about the conditions of biochemical
experiments, confidence of interactions, number of experiments confirming the
interactions, etc.
However, no other model for representing PPI data has been proposed
thus far. Despite all these drawbacks, understanding these complex phenomena is
crucial and can lead to significant discoveries about complex biological mechanisms
and diseases.
Fig : The PPI network with 11,000 interaction amongst 2401 yeast proteins

18. 18
4. Characterizing drug-drug target relationships:
An assessment of the number of drug targets, i.e., molecular targets
that represent an opportunity for therapeutic intervention, as well as their
identification, is crucial to the development of post-genomic research strategies
within the pharmaceutical industry.
Now that the size of the human genome is known, it is interesting to
consider just how many molecular targets this opportunity represents. Additionally,
identifying and characterizing the relationships between drugs and their protein
targets, as well as between drug targets and disease-gene products in the human
protein-protein interaction network still remains a challenge.
To study drug-drug target relationship, it will be divided into sub categories to study
drug-drug target relationship.
1. Drug able protein.
2. Drug Bank.
3. Drug Target Network.
Fig : Construction of diseasome bipartite network

19. 19
Application of Graph Theory in Operational Research:
Graph theory provides many useful applications in Operations
Research. A graph is defined as a finite number of points (known as nodes or
vertices) connected by lines (known as edges or arcs). In this paper for a given graph
find a minimum cost to find the shortest path between two points.
1. Minimum cost path:
From the Minimum Spanning Tree shown in Figure 1.1 we are able to
find the minimum cost path (trajectory) from node A to node B. As we can see from
the Figure 1.2, there are two alternative ways to reach from node A to node B, which
are distinguished by dash line.
Let’s start with first option to calculate the distance from node A to
node B (dash line), the result is as follows:
µ = 2+3+6+7+4+5+2+5+4+3+6+17 = 64 units (this is the most expensive path)
For the second option (full line): µ = 3+1+11+7+2 = 24 units
This means that the second option represents the minimum cost path
from node A to node B.
Fig 1.1 Fig 1.2

20. 20
2. A scheduling problem:
Suppose you're organizing a mathematics conference. You
have several people that are setting up special sessions in their area of
expertise. You're trying to find a way to schedule the sessions so that if
there are overlapping interests, you can make sure the sessions happen at
different times. To that end, you've asked each session organizer to tell
you what broad fields of mathematics their subject concerns. You have
the following list:
• Probabilistic Combinatorics: of interest to Analysts, Combinatorialists, and
Probabilists.
• Algebraic Combinatorics: of interest to Algebraists and Combinatorialists
• Enumerative Combinatorics: of interest to Combinatorialists and
Representation Theorists
• Ramsey Theory: of interest to Combinatorialists and Probabilists
• Markov Chains: of interest to Probabilists
• Brownian Motion: of interest to Probabilists and Analysts
• Operator Theory: of interest to Analysts
You'd like to schedule the shortest conference possible, so
that you don't have to pay for the rooms for too long. What is the fewest
number of sessions you could run to accommodate every session, where
no two topics run at the same time if they have overlapping interests?
Our scheduling problem can now be phrased as follows: divide the
vertices of the graph into sets so that no vertices with an edge between them belong
to the same set. This can be seen through the lens of graph coloring.
Figure 2.1: The Conflicts Graph for our Scheduling Program

21. 21
In Figure 2.2, we have a coloring of the conflicts graph that satisfies
the requirement we set. That is, no two vertices that are adjacent have the same
color. In this coloring, we have used four colors, which means we can assign our
seminars in four groups so that there are no conflicts. Specifically we have:
Fig 2.2: A Coloring of Conflicts Graph for Scheduling Problems

22. 22
Application of Graph Theory in Linguistics:
Graph-theoretic methods, in various forms, have proven particularly
useful in linguistics, since natural language often lends itself well to discrete
structure. Traditionally, syntax and compositional semantics follow tree-based
structures, whose expressive power lies in the principle of compositionality, modeled
in a hierarchical graph. More contemporary approaches such as head-driven phrase
structure grammar model the syntax of natural language using typed feature
structures, which are directed acyclic graphs.
Within lexical semantics, especially as applied to computers, modeling
word meaning is easier when a given word is understood in terms of related
words; semantic networks are therefore important in computational linguistics. Still
other methods in phonology (e.g. optimality theory, which uses lattice graphs) and
morphology (e.g. finite-state morphology, using finite-state transducers) are common
in the analysis of language as a graph. Indeed, the usefulness of this area of
mathematics to linguistics has borne organizations such as Text Graphs, as well as
various 'Net' projects, such as WorldNet, Verb Net, and others.
Application of Graph Theory in Physics:
Graph theory is also used to study molecules in physics. In condensed
matter physics, the three-dimensional structure of complicated simulated atomic
structures can be studied quantitatively by gathering statistics on graph-theoretic
properties related to the topology of the atoms.. In statistical physics, graphs can
represent local connections between interacting parts of a system, as well as the
dynamics of a physical process on such systems. Graphs are also used to represent
the micro-scale channels of porous media, in which the vertices represent the pores
and the edges represent the smaller channels connecting the pores.

23. 23
Application of Graph Theory in Sociology:
Graph theory is also widely used in sociology as a way, for example,
to measure actors' prestige or to explore rumor spreading, notably through the use
of social network analysis software. Under the umbrella of social networks are many
different types of graphs. Acquaintanceship and friendship graphs describe whether
people know each other. Influence graphs model whether certain people can
influence the behavior of others. Finally, collaboration graphs model whether two
people work together in a particular way, such as acting in a movie together.