introduction and representation of graph.
graph is collection of points and vertices.
there are 2 types of the graph 1. directed
2. undirected graph.
representation of graph 2 ways
Adjacency matrix
Adjacency list
On algorithmic problems concerning graphs of higher degree of symmetrygraphhoc
Since the ancient determination of the five platonic solids the study of symmetry and regularity has always
been one of the most fascinating aspects of mathematics. One intriguing phenomenon of studies in graph
theory is the fact that quite often arithmetic regularity properties of a graph imply the existence of many
symmetries, i.e. large automorphism group G. In some important special situation higher degree of
regularity means that G is an automorphism group of finite geometry. For example, a glance through the
list of distance regular graphs of diameter d < 3 reveals the fact that most of them are connected with
classical Lie geometry. Theory of distance regular graphs is an important part of algebraic combinatorics
and its applications such as coding theory, communication networks, and block design. An important tool
for investigation of such graphs is their spectra, which is the set of eigenvalues of adjacency matrix of a
graph. Let G be a finite simple group of Lie type and X be the set homogeneous elements of the associated
geometry. The complexity of computing the adjacency matrices of a graph Gr on the vertices X such that
Aut GR = G depends very much on the description of the geometry with which one starts. For example, we
can represent the geometry as the totality of 1 cosets of parabolic subgroups 2 chains of embedded
subspaces (case of linear groups), or totally isotropic subspaces (case of the remaining classical groups), 3
special subspaces of minimal module for G which are defined in terms of a G invariant multilinear form.
The aim of this research is to develop an effective method for generation of graphs connected with classical
geometry and evaluation of its spectra, which is the set of eigenvalues of adjacency matrix of a graph. The
main approach is to avoid manual drawing and to calculate graph layout automatically according to its
formal structure. This is a simple task in a case of a tree like graph with a strict hierarchy of entities but it
becomes more complicated for graphs of geometrical nature. There are two main reasons for the
investigations of spectra: (1) very often spectra carry much more useful information about the graph than a
corresponding list of entities and relationships (2) graphs with special spectra, satisfying so called
Ramanujan property or simply Ramanujan graphs (by name of Indian genius mathematician) are important
for real life applications (see [13]). There is a motivated suspicion that among geometrical graphs one
could find some new Ramanujan graphs.
ON ALGORITHMIC PROBLEMS CONCERNING GRAPHS OF HIGHER DEGREE OF SYMMETRYFransiskeran
Since the ancient determination of the five platonic solids the study of symmetry and regularity has always
been one of the most fascinating aspects of mathematics. One intriguing phenomenon of studies in graph
theory is the fact that quite often arithmetic regularity properties of a graph imply the existence of many
symmetries, i.e. large automorphism group G. In some important special situation higher degree of
regularity means that G is an automorphism group of finite geometry. For example, a glance through the
list of distance regular graphs of diameter d < 3 reveals the fact that most of them are connected with
classical Lie geometry. Theory of distance regular graphs is an important part of algebraic combinatorics
and its applications such as coding theory, communication networks, and block design. An important tool
for investigation of such graphs is their spectra, which is the set of eigenvalues of adjacency matrix of a
graph. Let G be a finite simple group of Lie type and X be the set homogeneous elements of the associated
geometry.
FREQUENT SUBGRAPH MINING ALGORITHMS - A SURVEY AND FRAMEWORK FOR CLASSIFICATIONcscpconf
Data mining algorithms are facing the challenge to deal with an increasing number of complex
objects. Graph is a natural data structure used for modeling complex objects. Frequent subgraph
mining is another active research topic in data mining . A graph is a general model to represent
data and has been used in many domains like cheminformatics and bioinformatics. Mining
patterns from graph databases is challenging since graph related operations, such as subgraph
testing, generally have higher time complexity than the corresponding operations on itemsets,
sequences, and trees. Many frequent subgraph Mining algorithms have been proposed. SPIN,
SUBDUE, g_Span, FFSM, GREW are a few to mention. In this paper we present a detailed
survey on frequent subgraph mining algorithms, which are used for knowledge discovery in
complex objects and also propose a frame work for classification of these algorithms. The
purpose is to help user to apply the techniques in a task specific manner in various application domains and to pave wave for further research.
introduction and representation of graph.
graph is collection of points and vertices.
there are 2 types of the graph 1. directed
2. undirected graph.
representation of graph 2 ways
Adjacency matrix
Adjacency list
On algorithmic problems concerning graphs of higher degree of symmetrygraphhoc
Since the ancient determination of the five platonic solids the study of symmetry and regularity has always
been one of the most fascinating aspects of mathematics. One intriguing phenomenon of studies in graph
theory is the fact that quite often arithmetic regularity properties of a graph imply the existence of many
symmetries, i.e. large automorphism group G. In some important special situation higher degree of
regularity means that G is an automorphism group of finite geometry. For example, a glance through the
list of distance regular graphs of diameter d < 3 reveals the fact that most of them are connected with
classical Lie geometry. Theory of distance regular graphs is an important part of algebraic combinatorics
and its applications such as coding theory, communication networks, and block design. An important tool
for investigation of such graphs is their spectra, which is the set of eigenvalues of adjacency matrix of a
graph. Let G be a finite simple group of Lie type and X be the set homogeneous elements of the associated
geometry. The complexity of computing the adjacency matrices of a graph Gr on the vertices X such that
Aut GR = G depends very much on the description of the geometry with which one starts. For example, we
can represent the geometry as the totality of 1 cosets of parabolic subgroups 2 chains of embedded
subspaces (case of linear groups), or totally isotropic subspaces (case of the remaining classical groups), 3
special subspaces of minimal module for G which are defined in terms of a G invariant multilinear form.
The aim of this research is to develop an effective method for generation of graphs connected with classical
geometry and evaluation of its spectra, which is the set of eigenvalues of adjacency matrix of a graph. The
main approach is to avoid manual drawing and to calculate graph layout automatically according to its
formal structure. This is a simple task in a case of a tree like graph with a strict hierarchy of entities but it
becomes more complicated for graphs of geometrical nature. There are two main reasons for the
investigations of spectra: (1) very often spectra carry much more useful information about the graph than a
corresponding list of entities and relationships (2) graphs with special spectra, satisfying so called
Ramanujan property or simply Ramanujan graphs (by name of Indian genius mathematician) are important
for real life applications (see [13]). There is a motivated suspicion that among geometrical graphs one
could find some new Ramanujan graphs.
ON ALGORITHMIC PROBLEMS CONCERNING GRAPHS OF HIGHER DEGREE OF SYMMETRYFransiskeran
Since the ancient determination of the five platonic solids the study of symmetry and regularity has always
been one of the most fascinating aspects of mathematics. One intriguing phenomenon of studies in graph
theory is the fact that quite often arithmetic regularity properties of a graph imply the existence of many
symmetries, i.e. large automorphism group G. In some important special situation higher degree of
regularity means that G is an automorphism group of finite geometry. For example, a glance through the
list of distance regular graphs of diameter d < 3 reveals the fact that most of them are connected with
classical Lie geometry. Theory of distance regular graphs is an important part of algebraic combinatorics
and its applications such as coding theory, communication networks, and block design. An important tool
for investigation of such graphs is their spectra, which is the set of eigenvalues of adjacency matrix of a
graph. Let G be a finite simple group of Lie type and X be the set homogeneous elements of the associated
geometry.
FREQUENT SUBGRAPH MINING ALGORITHMS - A SURVEY AND FRAMEWORK FOR CLASSIFICATIONcscpconf
Data mining algorithms are facing the challenge to deal with an increasing number of complex
objects. Graph is a natural data structure used for modeling complex objects. Frequent subgraph
mining is another active research topic in data mining . A graph is a general model to represent
data and has been used in many domains like cheminformatics and bioinformatics. Mining
patterns from graph databases is challenging since graph related operations, such as subgraph
testing, generally have higher time complexity than the corresponding operations on itemsets,
sequences, and trees. Many frequent subgraph Mining algorithms have been proposed. SPIN,
SUBDUE, g_Span, FFSM, GREW are a few to mention. In this paper we present a detailed
survey on frequent subgraph mining algorithms, which are used for knowledge discovery in
complex objects and also propose a frame work for classification of these algorithms. The
purpose is to help user to apply the techniques in a task specific manner in various application domains and to pave wave for further research.
The Roman Empire A Historical Colossus.pdfkaushalkr1407
The Roman Empire, a vast and enduring power, stands as one of history's most remarkable civilizations, leaving an indelible imprint on the world. It emerged from the Roman Republic, transitioning into an imperial powerhouse under the leadership of Augustus Caesar in 27 BCE. This transformation marked the beginning of an era defined by unprecedented territorial expansion, architectural marvels, and profound cultural influence.
The empire's roots lie in the city of Rome, founded, according to legend, by Romulus in 753 BCE. Over centuries, Rome evolved from a small settlement to a formidable republic, characterized by a complex political system with elected officials and checks on power. However, internal strife, class conflicts, and military ambitions paved the way for the end of the Republic. Julius Caesar’s dictatorship and subsequent assassination in 44 BCE created a power vacuum, leading to a civil war. Octavian, later Augustus, emerged victorious, heralding the Roman Empire’s birth.
Under Augustus, the empire experienced the Pax Romana, a 200-year period of relative peace and stability. Augustus reformed the military, established efficient administrative systems, and initiated grand construction projects. The empire's borders expanded, encompassing territories from Britain to Egypt and from Spain to the Euphrates. Roman legions, renowned for their discipline and engineering prowess, secured and maintained these vast territories, building roads, fortifications, and cities that facilitated control and integration.
The Roman Empire’s society was hierarchical, with a rigid class system. At the top were the patricians, wealthy elites who held significant political power. Below them were the plebeians, free citizens with limited political influence, and the vast numbers of slaves who formed the backbone of the economy. The family unit was central, governed by the paterfamilias, the male head who held absolute authority.
Culturally, the Romans were eclectic, absorbing and adapting elements from the civilizations they encountered, particularly the Greeks. Roman art, literature, and philosophy reflected this synthesis, creating a rich cultural tapestry. Latin, the Roman language, became the lingua franca of the Western world, influencing numerous modern languages.
Roman architecture and engineering achievements were monumental. They perfected the arch, vault, and dome, constructing enduring structures like the Colosseum, Pantheon, and aqueducts. These engineering marvels not only showcased Roman ingenuity but also served practical purposes, from public entertainment to water supply.
How to Make a Field invisible in Odoo 17Celine George
It is possible to hide or invisible some fields in odoo. Commonly using “invisible” attribute in the field definition to invisible the fields. This slide will show how to make a field invisible in odoo 17.
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Francesca Gottschalk from the OECD’s Centre for Educational Research and Innovation presents at the Ask an Expert Webinar: How can education support child empowerment?
Acetabularia Information For Class 9 .docxvaibhavrinwa19
Acetabularia acetabulum is a single-celled green alga that in its vegetative state is morphologically differentiated into a basal rhizoid and an axially elongated stalk, which bears whorls of branching hairs. The single diploid nucleus resides in the rhizoid.
Operation “Blue Star” is the only event in the history of Independent India where the state went into war with its own people. Even after about 40 years it is not clear if it was culmination of states anger over people of the region, a political game of power or start of dictatorial chapter in the democratic setup.
The people of Punjab felt alienated from main stream due to denial of their just demands during a long democratic struggle since independence. As it happen all over the word, it led to militant struggle with great loss of lives of military, police and civilian personnel. Killing of Indira Gandhi and massacre of innocent Sikhs in Delhi and other India cities was also associated with this movement.
A Strategic Approach: GenAI in EducationPeter Windle
Artificial Intelligence (AI) technologies such as Generative AI, Image Generators and Large Language Models have had a dramatic impact on teaching, learning and assessment over the past 18 months. The most immediate threat AI posed was to Academic Integrity with Higher Education Institutes (HEIs) focusing their efforts on combating the use of GenAI in assessment. Guidelines were developed for staff and students, policies put in place too. Innovative educators have forged paths in the use of Generative AI for teaching, learning and assessments leading to pockets of transformation springing up across HEIs, often with little or no top-down guidance, support or direction.
This Gasta posits a strategic approach to integrating AI into HEIs to prepare staff, students and the curriculum for an evolving world and workplace. We will highlight the advantages of working with these technologies beyond the realm of teaching, learning and assessment by considering prompt engineering skills, industry impact, curriculum changes, and the need for staff upskilling. In contrast, not engaging strategically with Generative AI poses risks, including falling behind peers, missed opportunities and failing to ensure our graduates remain employable. The rapid evolution of AI technologies necessitates a proactive and strategic approach if we are to remain relevant.
June 3, 2024 Anti-Semitism Letter Sent to MIT President Kornbluth and MIT Cor...Levi Shapiro
Letter from the Congress of the United States regarding Anti-Semitism sent June 3rd to MIT President Sally Kornbluth, MIT Corp Chair, Mark Gorenberg
Dear Dr. Kornbluth and Mr. Gorenberg,
The US House of Representatives is deeply concerned by ongoing and pervasive acts of antisemitic
harassment and intimidation at the Massachusetts Institute of Technology (MIT). Failing to act decisively to ensure a safe learning environment for all students would be a grave dereliction of your responsibilities as President of MIT and Chair of the MIT Corporation.
This Congress will not stand idly by and allow an environment hostile to Jewish students to persist. The House believes that your institution is in violation of Title VI of the Civil Rights Act, and the inability or
unwillingness to rectify this violation through action requires accountability.
Postsecondary education is a unique opportunity for students to learn and have their ideas and beliefs challenged. However, universities receiving hundreds of millions of federal funds annually have denied
students that opportunity and have been hijacked to become venues for the promotion of terrorism, antisemitic harassment and intimidation, unlawful encampments, and in some cases, assaults and riots.
The House of Representatives will not countenance the use of federal funds to indoctrinate students into hateful, antisemitic, anti-American supporters of terrorism. Investigations into campus antisemitism by the Committee on Education and the Workforce and the Committee on Ways and Means have been expanded into a Congress-wide probe across all relevant jurisdictions to address this national crisis. The undersigned Committees will conduct oversight into the use of federal funds at MIT and its learning environment under authorities granted to each Committee.
• The Committee on Education and the Workforce has been investigating your institution since December 7, 2023. The Committee has broad jurisdiction over postsecondary education, including its compliance with Title VI of the Civil Rights Act, campus safety concerns over disruptions to the learning environment, and the awarding of federal student aid under the Higher Education Act.
• The Committee on Oversight and Accountability is investigating the sources of funding and other support flowing to groups espousing pro-Hamas propaganda and engaged in antisemitic harassment and intimidation of students. The Committee on Oversight and Accountability is the principal oversight committee of the US House of Representatives and has broad authority to investigate “any matter” at “any time” under House Rule X.
• The Committee on Ways and Means has been investigating several universities since November 15, 2023, when the Committee held a hearing entitled From Ivory Towers to Dark Corners: Investigating the Nexus Between Antisemitism, Tax-Exempt Universities, and Terror Financing. The Committee followed the hearing with letters to those institutions on January 10, 202
Instructions for Submissions thorugh G- Classroom.pptxJheel Barad
This presentation provides a briefing on how to upload submissions and documents in Google Classroom. It was prepared as part of an orientation for new Sainik School in-service teacher trainees. As a training officer, my goal is to ensure that you are comfortable and proficient with this essential tool for managing assignments and fostering student engagement.
1. Page 21
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.
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INTERNATIONAL JOURNAL OF COMPUTER APPLICATION
ISSUE2, VOLUME 1 (FEBRUARY 2012) ISSN: 2250-1797
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.
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INTERNATIONAL JOURNAL OF COMPUTER APPLICATION
ISSUE2, VOLUME 1 (FEBRUARY 2012) ISSN: 2250-1797
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.
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INTERNATIONAL JOURNAL OF COMPUTER APPLICATION
ISSUE2, VOLUME 1 (FEBRUARY 2012) ISSN: 2250-1797
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.
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INTERNATIONAL JOURNAL OF COMPUTER APPLICATION
ISSUE2, VOLUME 1 (FEBRUARY 2012) ISSN: 2250-1797
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
6. Page 26
INTERNATIONAL JOURNAL OF COMPUTER APPLICATION
ISSUE2, VOLUME 1 (FEBRUARY 2012) ISSN: 2250-1797
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
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INTERNATIONAL JOURNAL OF COMPUTER APPLICATION
ISSUE2, VOLUME 1 (FEBRUARY 2012) ISSN: 2250-1797
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
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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.
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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.
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[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.