This document discusses various applications of graph theory to other areas of mathematics and other fields. It provides new graph theoretical proofs of Fermat's Little Theorem and the Nielsen-Schreier Theorem. It also discusses applications to problems in DNA sequencing, computer network security, scheduling, map coloring, and mobile phone networks. Specific algorithms for finding minimum vertex covers, vertex colorings, and matchings in graphs are applied to solve problems in these areas.
The document discusses graphs and their applications. It defines key graph terms like vertices, edges, directed/undirected graphs, paths, cycles, etc. It then describes algorithms for finding minimum spanning trees, Eulerian cycles, Hamiltonian paths, and approximations for the traveling salesman problem. Examples are provided to illustrate minimum spanning tree and Christofide's algorithms for TSP.
This document provides an overview of graph theory concepts including:
- The basics of graphs including definitions of vertices, edges, paths, cycles, and graph representations like adjacency matrices.
- Minimum spanning tree algorithms like Kruskal's and Prim's which find a spanning tree with minimum total edge weight.
- Graph coloring problems and their applications to scheduling problems.
- Other graph concepts covered include degree, Eulerian paths, planar graphs and graph isomorphism.
Graph theory is the study of points and lines, and concepts like vertices and edges. It has many applications, including map coloring, Google Maps, and football games. Graph theory can be used to color maps with the minimum number of colors without any adjacent regions sharing the same color. It is also used in Google Maps for functions like estimating travel times and showing traffic conditions. In football, a match can be represented as a network with players as vertices and passes as edges, allowing analysis of things like which team has a more balanced network.
This document provides an overview of graph theory and some of its common algorithms. It discusses the history of graph theory and its applications in various fields like engineering. It defines basic graph terminology like nodes, edges, walks, paths and cycles. It also explains popular graph algorithms like Dijkstra's algorithm for finding shortest paths, Kruskal's and Prim's algorithms for finding minimum spanning trees, and graph partitioning algorithms. It provides pseudocode, examples and analysis of the time complexity for these algorithms.
This document is a project report submitted by S. Manikanta in partial fulfillment of the requirements for a Master of Science degree in Mathematics. The report discusses applications of graph theory. It provides an overview of graph theory concepts such as definitions of graphs, terminology used in graph theory, different types of graphs, trees and forests, graph isomorphism and operations, walks and paths in graphs, representations of graphs using matrices, applications of graphs in areas like computer science, fingerprint recognition, security, and more. The document also includes examples and illustrations to explain various graph theory concepts.
The document provides an introduction to graph theory. It begins with a brief history, noting that graph theory originated from Euler's work on the Konigsberg bridges problem in 1735. It then discusses basic concepts such as graphs being collections of nodes and edges, different types of graphs like directed and undirected graphs. Key graph theory terms are defined such as vertices, edges, degree, walks, paths, cycles and shortest paths. Different graph representations like the adjacency matrix, incidence matrix and adjacency lists are also introduced. The document is intended to provide an overview of fundamental graph theory concepts.
The document provides an introduction to graph theory. It lists prescribed and recommended books, outlines topics that will be covered including history, definitions, types of graphs, terminology, representation, subgraphs, connectivity, and applications. It notes that the Government of India designated June 10th as Graph Theory Day in recognition of the influence and importance of graph theory.
The document discusses graphs and their applications. It defines key graph terms like vertices, edges, directed/undirected graphs, paths, cycles, etc. It then describes algorithms for finding minimum spanning trees, Eulerian cycles, Hamiltonian paths, and approximations for the traveling salesman problem. Examples are provided to illustrate minimum spanning tree and Christofide's algorithms for TSP.
This document provides an overview of graph theory concepts including:
- The basics of graphs including definitions of vertices, edges, paths, cycles, and graph representations like adjacency matrices.
- Minimum spanning tree algorithms like Kruskal's and Prim's which find a spanning tree with minimum total edge weight.
- Graph coloring problems and their applications to scheduling problems.
- Other graph concepts covered include degree, Eulerian paths, planar graphs and graph isomorphism.
Graph theory is the study of points and lines, and concepts like vertices and edges. It has many applications, including map coloring, Google Maps, and football games. Graph theory can be used to color maps with the minimum number of colors without any adjacent regions sharing the same color. It is also used in Google Maps for functions like estimating travel times and showing traffic conditions. In football, a match can be represented as a network with players as vertices and passes as edges, allowing analysis of things like which team has a more balanced network.
This document provides an overview of graph theory and some of its common algorithms. It discusses the history of graph theory and its applications in various fields like engineering. It defines basic graph terminology like nodes, edges, walks, paths and cycles. It also explains popular graph algorithms like Dijkstra's algorithm for finding shortest paths, Kruskal's and Prim's algorithms for finding minimum spanning trees, and graph partitioning algorithms. It provides pseudocode, examples and analysis of the time complexity for these algorithms.
This document is a project report submitted by S. Manikanta in partial fulfillment of the requirements for a Master of Science degree in Mathematics. The report discusses applications of graph theory. It provides an overview of graph theory concepts such as definitions of graphs, terminology used in graph theory, different types of graphs, trees and forests, graph isomorphism and operations, walks and paths in graphs, representations of graphs using matrices, applications of graphs in areas like computer science, fingerprint recognition, security, and more. The document also includes examples and illustrations to explain various graph theory concepts.
The document provides an introduction to graph theory. It begins with a brief history, noting that graph theory originated from Euler's work on the Konigsberg bridges problem in 1735. It then discusses basic concepts such as graphs being collections of nodes and edges, different types of graphs like directed and undirected graphs. Key graph theory terms are defined such as vertices, edges, degree, walks, paths, cycles and shortest paths. Different graph representations like the adjacency matrix, incidence matrix and adjacency lists are also introduced. The document is intended to provide an overview of fundamental graph theory concepts.
The document provides an introduction to graph theory. It lists prescribed and recommended books, outlines topics that will be covered including history, definitions, types of graphs, terminology, representation, subgraphs, connectivity, and applications. It notes that the Government of India designated June 10th as Graph Theory Day in recognition of the influence and importance of graph theory.
This document introduces the topic of graph theory. It defines what graphs are, including vertices, edges, directed and undirected graphs. It provides examples of graphs like social networks, transportation maps, and more. It covers basic graph terminology such as degree, regular graphs, subgraphs, walks, paths and cycles. It also discusses graph classes like trees, complete graphs and bipartite graphs. Finally, it touches on some historical graph problems, complexity analysis, centrality analysis, facility location problems and applications of graph theory.
In mathematics, graph theory is the study of graphs, which are mathematical structures used to model pairwise relations between objects.Graph theory is also important in real life.
Graph theory is widely used in science and everyday life. It can model real world problems and systems using vertices to represent objects and edges to represent connections between objects. The document discusses several applications of graph theory in chemistry, physics, biology, computer science, operations research, Google Maps, and the internet. For example, in chemistry graph theory is used to model molecules with atoms as vertices and bonds as edges. In computer science, graph theory concepts are used to develop algorithms for problems like finding shortest paths in a network.
This document introduces graph theory and provides examples of graphs in the real world. It discusses how graphs are used to represent connections between objects and discusses some key graph concepts like nodes, edges, paths, and degrees. Real-world examples of graphs mentioned include social networks, maps, and the structure of the internet. The document also explains why graph theory is useful for modeling real-world networks and solving optimization problems.
This document provides an overview of graph theory, including its history and key concepts. It defines a graph as a set of nodes connected by edges. Important early contributors are noted, such as Euler's work on the Seven Bridges of Königsberg problem. Exact definitions of graph types like simple, directed, and weighted graphs are given. Key graph concepts explained in less than three sentences include connectivity, degree, paths, cycles, and trees. The document also briefly discusses representing graphs through adjacency lists and matrices. It concludes by thanking the reader and listing references.
This document presents information about graph coloring. It defines graph coloring as assigning colors to the vertices of a graph so that no adjacent vertices have the same color. There are two types: edge coloring and vertex coloring. Graph coloring is useful for problems like scheduling and radio channel assignment. However, finding the minimum number of colors to color a graph is an NP-complete problem, meaning there is no known efficient algorithm. Some common graph coloring algorithms discussed are greedy algorithm and Welsh-Powell algorithm.
This section provides an introduction to graphs and graph theory. Key points include:
- Graphs consist of vertices and edges that connect the vertices. They can be directed or undirected.
- Common terminology is introduced, such as adjacent vertices, neighborhoods, degrees of vertices, and handshaking theorem.
- Different types of graphs are discussed, including multigraphs, pseudographs, and directed graphs.
- Examples of graph models are given for computer networks, social networks, information networks, transportation networks, and software design. Graphs can be used to model many real-world systems and applications.
This document discusses graph coloring and bipartite graphs. It defines a bipartite graph as one whose nodes can be partitioned into two sets such that no two nodes within the same set are adjacent. Bipartite graphs can be colored with two colors by coloring all nodes in one set one color and nodes in the other set the other color. The four color theorem states that any separation of a plane into contiguous regions can be colored with no more than four colors such that no two adjacent regions have the same color. Applications of graph coloring include exam scheduling, radio frequency assignment, register allocation, map coloring, and Sudoku puzzles.
Graph theory is the study of graphs, which are mathematical structures used to model pairwise relationships between objects. A graph consists of vertices and edges that connect the vertices. Graph theory is used in computer science to model problems that can be represented as networks, such as determining routes in a city or designing circuits. It allows analyzing properties of graphs like connectivity and determining the minimum number of resources required. A graph can be represented using an adjacency matrix or adjacency list to store information about its vertices and edges.
This document discusses graph coloring and its applications. It begins by defining graph coloring as assigning labels or colors to elements of a graph such that no two adjacent elements have the same color. It then provides examples of vertex coloring, edge coloring, and face coloring. The document also discusses the chromatic number and chromatic polynomial. It describes several real-world applications of graph coloring, including frequency assignment in cellular networks.
Graph theory concepts complex networks presents-rouhollah nabatinabati
This document provides an introduction to network and social network analysis theory, including basic concepts of graph theory and network structures. It defines what a network and graph are, explains what network theory techniques are used for, and gives examples of real-world networks that can be represented as graphs. It also summarizes key graph theory concepts such as nodes, edges, walks, paths, cycles, connectedness, degree, and centrality measures.
This document provides an introduction to graph theory through a presentation on the topic. It defines what a graph is by explaining that a graph G consists of a set of vertices V and edges E. It then gives examples and defines basic terminology like adjacency and incidence. The document also covers topics like degrees of vertices, regular and bipartite graphs, and representations of graphs through adjacency and incidence matrices.
The document discusses various graph theory topics including isomorphism, cut sets, labeled graphs, and Hamiltonian circuits. It defines isomorphism as two graphs being structurally identical with a one-to-one correspondence between their vertices and edges. Cut sets are edges whose removal would disconnect a connected graph. Labeled graphs assign labels or weights to their vertices and/or edges. A Hamiltonian circuit is a closed walk that visits each vertex exactly once.
Graph isomorphism is the problem of determining whether two graphs are topologically identical. It is in NP but not known to be NP-complete. The refinement heuristic iteratively partitions vertices into equivalence classes to reduce possible mappings between graphs. For trees, there is a linear-time algorithm that labels vertices based on subtree structure to test rooted and unrooted tree isomorphism.
Application of graph theory in Traffic ManagementSwathiSundari
Traffic congestion has become a major problem in many urban areas. Constructing new roads is very costly and often not possible. The most efficient way to manage traffic is through proper traffic light timing and phasing. The document describes a methodology to model traffic flow at a junction as a graph and use linear programming to determine optimal light phasing durations to minimize total wait time.
Graph theory is the study of points and lines, and how sets of points called vertices can be connected by lines called edges. It involves types of graphs like regular graphs where each vertex has the same number of neighbors, and bipartite graphs where the vertices can be partitioned into two sets with no edges within each set. Graphs can be represented using adjacency matrices and adjacency lists. Basic graph algorithms include depth-first search, breadth-first search, and finding shortest paths between vertices. Graph coloring assigns colors to vertices so that no adjacent vertices have the same color.
Graph theory is a branch of mathematics that studies networks and their structures. It began in 1736 when Euler solved the Konigsberg bridges problem. Graphs represent objects called nodes connected by relationships called edges. They are used to model many real-world applications. Similarity searching uses graph representations of molecules to identify structurally or pharmacologically similar compounds. Various molecular descriptors and similarity coefficients are used with substructure, reduced graph, and 3D methods. Data fusion combines results from multiple search methods to improve accuracy. Graph theory continues to be important for modeling complex systems and networks in science.
Graph theory has many applications including social networks, data organization, and communication networks. The document discusses Dijkstra's algorithm for finding the shortest path between nodes in a graph and its application to finding shortest routes between cities. It also discusses using graph representations for fingerprint classification, where fingerprints are modeled as graphs with nodes for fingerprint regions and edges between adjacent regions. Fingerprints are classified based on the structure of these graphs and compared to model graphs for matching.
This document introduces the topic of graph theory. It defines what graphs are, including vertices, edges, directed and undirected graphs. It provides examples of graphs like social networks, transportation maps, and more. It covers basic graph terminology such as degree, regular graphs, subgraphs, walks, paths and cycles. It also discusses graph classes like trees, complete graphs and bipartite graphs. Finally, it touches on some historical graph problems, complexity analysis, centrality analysis, facility location problems and applications of graph theory.
In mathematics, graph theory is the study of graphs, which are mathematical structures used to model pairwise relations between objects.Graph theory is also important in real life.
Graph theory is widely used in science and everyday life. It can model real world problems and systems using vertices to represent objects and edges to represent connections between objects. The document discusses several applications of graph theory in chemistry, physics, biology, computer science, operations research, Google Maps, and the internet. For example, in chemistry graph theory is used to model molecules with atoms as vertices and bonds as edges. In computer science, graph theory concepts are used to develop algorithms for problems like finding shortest paths in a network.
This document introduces graph theory and provides examples of graphs in the real world. It discusses how graphs are used to represent connections between objects and discusses some key graph concepts like nodes, edges, paths, and degrees. Real-world examples of graphs mentioned include social networks, maps, and the structure of the internet. The document also explains why graph theory is useful for modeling real-world networks and solving optimization problems.
This document provides an overview of graph theory, including its history and key concepts. It defines a graph as a set of nodes connected by edges. Important early contributors are noted, such as Euler's work on the Seven Bridges of Königsberg problem. Exact definitions of graph types like simple, directed, and weighted graphs are given. Key graph concepts explained in less than three sentences include connectivity, degree, paths, cycles, and trees. The document also briefly discusses representing graphs through adjacency lists and matrices. It concludes by thanking the reader and listing references.
This document presents information about graph coloring. It defines graph coloring as assigning colors to the vertices of a graph so that no adjacent vertices have the same color. There are two types: edge coloring and vertex coloring. Graph coloring is useful for problems like scheduling and radio channel assignment. However, finding the minimum number of colors to color a graph is an NP-complete problem, meaning there is no known efficient algorithm. Some common graph coloring algorithms discussed are greedy algorithm and Welsh-Powell algorithm.
This section provides an introduction to graphs and graph theory. Key points include:
- Graphs consist of vertices and edges that connect the vertices. They can be directed or undirected.
- Common terminology is introduced, such as adjacent vertices, neighborhoods, degrees of vertices, and handshaking theorem.
- Different types of graphs are discussed, including multigraphs, pseudographs, and directed graphs.
- Examples of graph models are given for computer networks, social networks, information networks, transportation networks, and software design. Graphs can be used to model many real-world systems and applications.
This document discusses graph coloring and bipartite graphs. It defines a bipartite graph as one whose nodes can be partitioned into two sets such that no two nodes within the same set are adjacent. Bipartite graphs can be colored with two colors by coloring all nodes in one set one color and nodes in the other set the other color. The four color theorem states that any separation of a plane into contiguous regions can be colored with no more than four colors such that no two adjacent regions have the same color. Applications of graph coloring include exam scheduling, radio frequency assignment, register allocation, map coloring, and Sudoku puzzles.
Graph theory is the study of graphs, which are mathematical structures used to model pairwise relationships between objects. A graph consists of vertices and edges that connect the vertices. Graph theory is used in computer science to model problems that can be represented as networks, such as determining routes in a city or designing circuits. It allows analyzing properties of graphs like connectivity and determining the minimum number of resources required. A graph can be represented using an adjacency matrix or adjacency list to store information about its vertices and edges.
This document discusses graph coloring and its applications. It begins by defining graph coloring as assigning labels or colors to elements of a graph such that no two adjacent elements have the same color. It then provides examples of vertex coloring, edge coloring, and face coloring. The document also discusses the chromatic number and chromatic polynomial. It describes several real-world applications of graph coloring, including frequency assignment in cellular networks.
Graph theory concepts complex networks presents-rouhollah nabatinabati
This document provides an introduction to network and social network analysis theory, including basic concepts of graph theory and network structures. It defines what a network and graph are, explains what network theory techniques are used for, and gives examples of real-world networks that can be represented as graphs. It also summarizes key graph theory concepts such as nodes, edges, walks, paths, cycles, connectedness, degree, and centrality measures.
This document provides an introduction to graph theory through a presentation on the topic. It defines what a graph is by explaining that a graph G consists of a set of vertices V and edges E. It then gives examples and defines basic terminology like adjacency and incidence. The document also covers topics like degrees of vertices, regular and bipartite graphs, and representations of graphs through adjacency and incidence matrices.
The document discusses various graph theory topics including isomorphism, cut sets, labeled graphs, and Hamiltonian circuits. It defines isomorphism as two graphs being structurally identical with a one-to-one correspondence between their vertices and edges. Cut sets are edges whose removal would disconnect a connected graph. Labeled graphs assign labels or weights to their vertices and/or edges. A Hamiltonian circuit is a closed walk that visits each vertex exactly once.
Graph isomorphism is the problem of determining whether two graphs are topologically identical. It is in NP but not known to be NP-complete. The refinement heuristic iteratively partitions vertices into equivalence classes to reduce possible mappings between graphs. For trees, there is a linear-time algorithm that labels vertices based on subtree structure to test rooted and unrooted tree isomorphism.
Application of graph theory in Traffic ManagementSwathiSundari
Traffic congestion has become a major problem in many urban areas. Constructing new roads is very costly and often not possible. The most efficient way to manage traffic is through proper traffic light timing and phasing. The document describes a methodology to model traffic flow at a junction as a graph and use linear programming to determine optimal light phasing durations to minimize total wait time.
Graph theory is the study of points and lines, and how sets of points called vertices can be connected by lines called edges. It involves types of graphs like regular graphs where each vertex has the same number of neighbors, and bipartite graphs where the vertices can be partitioned into two sets with no edges within each set. Graphs can be represented using adjacency matrices and adjacency lists. Basic graph algorithms include depth-first search, breadth-first search, and finding shortest paths between vertices. Graph coloring assigns colors to vertices so that no adjacent vertices have the same color.
Graph theory is a branch of mathematics that studies networks and their structures. It began in 1736 when Euler solved the Konigsberg bridges problem. Graphs represent objects called nodes connected by relationships called edges. They are used to model many real-world applications. Similarity searching uses graph representations of molecules to identify structurally or pharmacologically similar compounds. Various molecular descriptors and similarity coefficients are used with substructure, reduced graph, and 3D methods. Data fusion combines results from multiple search methods to improve accuracy. Graph theory continues to be important for modeling complex systems and networks in science.
Graph theory has many applications including social networks, data organization, and communication networks. The document discusses Dijkstra's algorithm for finding the shortest path between nodes in a graph and its application to finding shortest routes between cities. It also discusses using graph representations for fingerprint classification, where fingerprints are modeled as graphs with nodes for fingerprint regions and edges between adjacent regions. Fingerprints are classified based on the structure of these graphs and compared to model graphs for matching.
Talk given at neo4j conference "Graph Connect" - discussing some graph theory (old and new), and why knowing your stuff can come in handy on a software project.
1. The document discusses how graph theory concepts like degree centrality, betweenness centrality, and center were applied to analyze football matches from the 2010 FIFA World Cup semi-finals and final.
2. By constructing graphs based on pass networks, it was able to predict that Spain would defeat the Netherlands in the final based on Spain having a more well-balanced and interconnected passing strategy compared to the Netherlands' more predictable attack.
3. Spain's graph showed low, evenly distributed betweenness scores, indicating a well-balanced passing game, while the Netherlands' scores were more concentrated, showing dependency on few key players. The analysis confirmed Spain played a clever game relying on many short passes between a network of midfielders.
the bike map - a look into a practical application of graph theoryCharlie Hsu
The document describes how a bike route map was created using graph theory and algorithms. Key nodes (intersections) and edges (street segments) were generated from public transit and map data. An A* search algorithm was then used to find optimal routes between locations by minimizing a cost function based on distance and elevation change. The cost function applies penalties for steep grades and road types to encourage bike-friendly routes.
Graph coloring involves assigning colors to objects in a graph subject to constraints. Vertex coloring assigns colors to vertices such that no adjacent vertices share the same color. The chromatic number is the minimum number of colors needed. Graph coloring has applications in scheduling, frequency assignment, and register allocation. Register allocation with chordal graphs can be done in polynomial time rather than NP-complete, since programs in SSA form have chordal interference graphs. A greedy algorithm can color chordal graphs in linear time, enabling simple and optimal register allocation.
This document provides definitions and theorems related to graph theory. It begins with definitions of simple graphs, vertices, edges, degree, and the handshaking lemma. It then covers definitions and properties of paths, cycles, adjacency matrices, connectedness, Euler paths and circuits. The document also discusses Hamilton paths, planar graphs, trees, and other special types of graphs like complete graphs and bipartite graphs. It provides examples and proofs of many graph theory concepts and results.
The document discusses applications of graph theory beyond engineering problems. It provides an introduction to graph theory and its history from Leonhard Euler's Seven Bridges of Königsberg problem in 1736. Examples of using graph theory include representing relationships with graphs, drawing a house with one line, coloring a graph to arrange students' seating to prevent copying, and using path search algorithms in complex systems like aircraft flight control. Graph theory can model many real-world problems and using appropriate graph algorithms can provide benefits.
The document provides an overview of graph theory and applications. It begins with a brief history of graph theory starting with Euler and Hamilton. It then summarizes some key graph theory concepts like connectivity, paths, trees, and coloring problems. The document outlines several applications of graph theory including ranking web pages, finding the shortest path with GPS, and analyzing large networks and graphs. It concludes by mentioning some large scale graph problems like similarity of nodes, telephony networks, and clustering large graphs.
1) The document introduces basic concepts in graph theory including definitions of graphs, vertices, edges and examples.
2) It discusses some classic problems in graph theory including Euler's solving of the Seven Bridges of Königsberg problem, which helped formulate graph theory.
3) The document defines Euler paths, Euler circuits and the conditions for when a graph contains them based on the number of odd and even vertices.
The document provides an introduction to graph theory concepts. It defines graphs as pairs of vertices and edges, and introduces basic graph terminology like order, size, adjacency, and isomorphism. Graphs can be represented geometrically by drawing vertices as points and edges as lines between them. Both simple graphs and multigraphs are discussed.
This slide was presented by the Maths Department of Cochin Refineries School for the Inter-School workshop conducted as a part of World Mathematics Day celebration. "Mathematics in day to day life"
This document introduces discrete mathematics and its applications. It discusses how discrete mathematics underlies computer science and problem solving. Discrete structures are used to represent and manipulate digital data. The document outlines the main topics covered in a discrete mathematics course, including sets, relations, groups, graphs, trees, and discrete probability. It provides examples of how these concepts are applied to domains like social networks, software design, and routing algorithms.
This document provides details on a new software update that will be installed on all company computers. The update includes security patches that fix vulnerabilities, improved compatibility with newer operating systems, and new features to enhance the user experience. The update will be automatically pushed out to all devices overnight on Friday and should take less than 30 minutes to complete on most machines. Users are advised to make sure their computers have a backup battery or are plugged in to avoid power interruptions during installation.
This document outlines research on applying graph theory to solve the vehicle routing problem. It defines a graph and the vehicle routing problem, poses the research question of identifying solution techniques for solving the vehicle routing problem, and puts forth the hypothesis that these techniques can be matched to a graph model. The document previews support from various sources and references and notes the author's favorite related topic.
Cs6702 graph theory and applications question bankappasami
This document contains questions and answers related to the course CS6702 Graph Theory and Applications. It is divided into 6 units which cover topics like introduction to graphs, trees, connectivity, planarity, graph colorings, directed graphs, permutations and combinations, and generating functions. For each unit, it lists short answer questions in Part A and longer proof or explanation questions in Part B. The questions test concepts fundamental to graph theory such as definitions of graphs, trees, connectivity, colorings, isomorphism, and counting methods applied to graph theory.
Introduction to graph of class 8th students. Find a new easy way to understand graph, histogram, double-bar graph, pie-chart etc....This ppt could lead to u a better picture of maths
This document provides an introduction to fractal geometry and its applications. It discusses how fractals were discovered by Mandelbrot as a way to describe irregular patterns in nature. Some key areas where fractals are applied that are mentioned include astronomy (galaxies and Saturn's rings), biology (bacteria cultures, plants), data compression, special effects in movies like Star Trek, weather patterns, heartbeats, antennas, and modeling global warming. The document also provides mathematical definitions of fractals and the Mandelbrot set, and describes how to generate fractal landscapes and the Mandelbrot set using iterated functions.
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.
THE RESULT FOR THE GRUNDY NUMBER ON P4- CLASSESgraphhoc
This document summarizes research on calculating the Grundy number of fat-extended P4-laden graphs. It begins by introducing the Grundy number and discussing that it is NP-complete to calculate for general graphs. It then presents previous work that has found polynomial time algorithms to calculate the Grundy number for certain graph classes. The main results are that the document proves that the Grundy number can be calculated in polynomial time, specifically O(n3) time, for fat-extended P4-laden graphs by traversing their modular decomposition tree. This implies the Grundy number can also be calculated efficiently for several related graph classes that are contained within fat-extended P4-laden graphs.
The document discusses the planted clique problem in graph theory. It introduces the problem and describes how previous research has found polynomial-time algorithms to solve the problem when the size of the planted clique k is O(√n). The document then summarizes two algorithms - Kucera's algorithm and the Low Degree Removal (LDR) algorithm - that have been used to approach the problem. It describes implementing the algorithms in a C++ program to simulate random graphs with planted cliques and test the ability of the algorithms to recover the planted clique.
Graph Dynamical System on Graph ColouringClyde Shen
This document discusses using a graph dynamical system to model the vertex coloring problem. It proposes using a Graph-cellular Automaton (GA), which is an extension of cellular automata, to distribute the coloring process. In the GA model, each vertex independently chooses its optimal color based on the colors of its adjacent vertices. The document outlines testing the GA approach on different types of graphs and analyzing whether it can find colorings for them. It provides background on graph coloring, planar graphs, and introduces the graph dynamical system and GA models.
Cs6660 compiler design november december 2016 Answer keyappasami
The document discusses topics related to compiler design, including:
1) The phases of a compiler include lexical analysis, syntax analysis, semantic analysis, intermediate code generation, code optimization, and code generation. Compiler construction tools help implement these phases.
2) Grouping compiler phases can improve efficiency. Errors can occur in all phases, from syntax errors to type errors.
3) Questions cover topics like symbol tables, finite automata in lexical analysis, parse trees, ambiguity, SLR parsing, syntax directed translations, code generation, and optimization techniques like loop detection.
This document summarizes applications of graph theory concepts. It discusses using minimum spanning trees for network design problems to connect offices at minimum cost. It also discusses using bipartite graphs and matchings to model job assignment problems, and using graph coloring to create exam schedules where no two exams with a common student occur at the same time. Other applications mentioned include using planar graphs in circuit board design and transportation planning.
A NEW PARALLEL ALGORITHM FOR COMPUTING MINIMUM SPANNING TREEijscmc
Computing the minimum spanning tree of the graph is one of the fundamental computational problems. In
this paper, we present a new parallel algorithm for computing the minimum spanning tree of an undirected
weighted graph with n vertices and m edges. This algorithm uses the cluster techniques to reduce the
number of processors by fraction 1/f (n) and the parallel work by the fraction O ( 1 lo g ( f ( n )) ),where f (n) is an
arbitrary function. In the case f (n) =1, the algorithm runs in logarithmic-time and use super linear work on
EREWPRAM model. In general, the proposed algorithm is the simplest one.
A NEW PARALLEL ALGORITHM FOR COMPUTING MINIMUM SPANNING TREEijscmcj
Computing the minimum spanning tree of the graph is one of the fundamental computational problems. In this paper, we present a new parallel algorithm for computing the minimum spanning tree of an undirected weighted graph with vertices and edges. This algorithm uses the cluster techniques to reduce the number of processors by fraction and the parallel work by the fraction O ( 1 lo g ( f ( n )) ),where f (n) is an arbitrary function. In the case f (n) =1, the algorithm runs in logarithmic-time and use super linear work on EREWPRAM model. In general, the proposed algorithm is the simplest one.
In many scientific areas, systems can be described as interaction networks where elements correspond to vertices and interactions to edges. A variety of problems in those fields can deal with network comparison and characterization.
The problem of comparing and characterizing networks is the task of measuring their structural similarity and finding characteristics which capture structural information. In order to analyze complex networks, several methods can be combined, such as graph theory, information theory, and statistics.
In this project, we present methods for measuring Shannon’s entropy of graphs.
A Quest for Subexponential Time Parameterized Algorithms for Planar-k-Path: F...cseiitgn
The document summarizes a talk on obtaining subexponential time algorithms for NP-hard problems on planar graphs. It discusses using treewidth and tree decompositions to solve problems like 3-coloring in 2O(√n) time on n-vertex planar graphs. It also discusses the exponential time hypothesis and how it implies lower bounds, showing these algorithms are optimal up to constant factors in the exponent. The document outlines several chapters, including using grid minors and bidimensionality to obtain 2O(√k) algorithms for problems like k-path, even for some W[1]-hard problems parameterized by k.
Average Polynomial Time Complexity Of Some NP-Complete ProblemsAudrey Britton
This document discusses the average polynomial time complexity of some NP-complete problems. Specifically:
- It proves that when N(n) = [n^e] for 0 < e < 2, the CLIQUEN(n) problem (finding a clique of size k in a graph with n vertices and at most N(n) edges) is NP-complete, yet can be solved in average and almost everywhere polynomial time.
- It considers the CLIQUE problem under different non-uniform distributions on the input graphs, showing the problem can be solved in average polynomial time for certain values of the distribution parameter p.
- The main result shows that a family of subproblems of CLIQUE that remain
Algorithm for Edge Antimagic Labeling for Specific Classes of GraphsCSCJournals
Graph labeling is a remarkable field having direct and indirect involvement in resolving numerous issues in varied fields. During this paper we tend to planned new algorithms to construct edge antimagic vertex labeling, edge antimagic total labeling, (a, d)-edge antimagic vertex labeling, (a, d)-edge antimagic total labeling and super edge antimagic labeling of varied classes of graphs like paths, cycles, wheels, fans, friendship graphs. With these solutions several open issues during this space can be solved.
11.characterization of connected vertex magic total labeling graphs in topolo...Alexander Decker
The document summarizes research characterizing connected vertex magic total labeling (VMTL) graphs through ideals in topological spaces. Some key results:
1. A connected VMTL graph G is Eulerian if and only if all vertices of G have even degree.
2. For a connected VMTL graph G, topology τ on vertices, and ideal I of even degree vertices, a vertex v has empty local function if and only if its degree is even.
3. G is Eulerian if and only if all local functions are empty, characterizing even degree vertices through ideals in topological spaces.
4. [22 25]characterization of connected vertex magic total labeling graphs in...Alexander Decker
The document summarizes research characterizing connected vertex magic total labeling (VMTL) graphs through ideals in topological spaces. Some key results:
1. A connected VMTL graph G is Eulerian if and only if all vertices of G have even degree.
2. For a connected VMTL graph G, ideal I of even degree vertices, and topology τ on vertices, the local function {v}* is empty if and only if the degree of v is even.
3. G is Eulerian if and only if the local function is empty for all vertices, characterizing when G has only even degree vertices.
The paper presents theorems relating the topological concept of local functions of
4. [22 25]characterization of connected vertex magic total labeling graphs in...Alexander Decker
The document summarizes research characterizing connected vertex magic total labeling (VMTL) graphs through ideals in topological spaces. Some key results:
1. A connected VMTL graph G is Eulerian if and only if all vertices of G have even degree.
2. For a connected VMTL graph G, ideal I of even degree vertices, and topology τ on vertices, the local function {v}* is empty if and only if the degree of v is even.
3. G is Eulerian if and only if the local function is empty for all vertices, characterizing when G has only even degree vertices.
The paper presents theorems relating the topological concept of local functions of
Elliptic Curves as Tool for Public Key Cryptographyinventy
Research Inventy : International Journal of Engineering and Science is published by the group of young academic and industrial researchers with 12 Issues per year. It is an online as well as print version open access journal that provides rapid publication (monthly) of articles in all areas of the subject such as: civil, mechanical, chemical, electronic and computer engineering as well as production and information technology. The Journal welcomes the submission of manuscripts that meet the general criteria of significance and scientific excellence. Papers will be published by rapid process within 20 days after acceptance and peer review process takes only 7 days. All articles published in Research Inventy will be peer-reviewed.
A Branch And Bound Algorithm For The Maximum Clique ProblemSara Alvarez
This document presents a branch and bound algorithm for solving the maximum clique problem based on modeling it as an equivalent quadratic zero-one programming formulation. The algorithm uses dynamic variable selection to order the branching tree. Two different rules for selecting vertices/variables are compared: 1) a greedy approach that selects high connectivity vertices first, and 2) a non-greedy approach that selects low connectivity vertices first. While the greedy approach finds a maximum clique solution sooner, the non-greedy approach typically verifies optimality with a smaller search tree. Computational results on randomly generated graphs with up to 1000 vertices and 150,000 edges demonstrate the effectiveness of the algorithms.
The document discusses graphs and graph algorithms. It defines what a graph is - a non-linear data structure consisting of vertices connected by edges. It describes different graph representations like adjacency matrix, incidence matrix and adjacency list. It also explains graph traversal algorithms like depth-first search and breadth-first search. Finally, it covers minimum spanning tree algorithms like Kruskal's and Prim's algorithms for finding minimum cost spanning trees in graphs.
A review of the growth of the Israel Genealogy Research Association Database Collection for the last 12 months. Our collection is now passed the 3 million mark and still growing. See which archives have contributed the most. See the different types of records we have, and which years have had records added. You can also see what we have for the future.
हिंदी वर्णमाला पीपीटी, hindi alphabet PPT presentation, hindi varnamala PPT, Hindi Varnamala pdf, हिंदी स्वर, हिंदी व्यंजन, sikhiye hindi varnmala, dr. mulla adam ali, hindi language and literature, hindi alphabet with drawing, hindi alphabet pdf, hindi varnamala for childrens, hindi language, hindi varnamala practice for kids, https://www.drmullaadamali.com
How to Make a Field Mandatory in Odoo 17Celine George
In Odoo, making a field required can be done through both Python code and XML views. When you set the required attribute to True in Python code, it makes the field required across all views where it's used. Conversely, when you set the required attribute in XML views, it makes the field required only in the context of that particular view.
Main Java[All of the Base Concepts}.docxadhitya5119
This is part 1 of my Java Learning Journey. This Contains Custom methods, classes, constructors, packages, multithreading , try- catch block, finally block and more.
How to Setup Warehouse & Location in Odoo 17 InventoryCeline George
In this slide, we'll explore how to set up warehouses and locations in Odoo 17 Inventory. This will help us manage our stock effectively, track inventory levels, and streamline warehouse operations.
Executive Directors Chat Leveraging AI for Diversity, Equity, and InclusionTechSoup
Let’s explore the intersection of technology and equity in the final session of our DEI series. Discover how AI tools, like ChatGPT, can be used to support and enhance your nonprofit's DEI initiatives. Participants will gain insights into practical AI applications and get tips for leveraging technology to advance their DEI goals.
Executive Directors Chat Leveraging AI for Diversity, Equity, and Inclusion
Graph theory and life
1. GRAPHTHEORY AND lifeBy MILAN. A.JOSHI (H.O.D) & PROFESSOR DEPARTMENTOF MATHEMATICS BHONSLA COLLEGE OF ENGINEERING AND RESEARCH
2. ABSTRACT Graph theory is becoming increasingly significant as it is applied to other areas of mathematics, science and technology. It is being actively used in fields as varied as biochemistry (genomics), electrical engineering (communication networks and coding theory), computer science (algorithms and computation) and operations research (scheduling). The powerful combinatorial methods found in graph theory have also been used to prove fundamental results in other areas of pure mathematics. This paper, besides giving a general outlook of these facts, includes new graph theoretical proofs of Fermat’s Little theorem and the Nielson-Schreier’s theorem. New applications to DNAsequencing (the SNP assembly problem) and computer network security (worm propagation) using minimum vertex covers in graphs are discussed. We also show how to apply edge coloring and matching in graphs for scheduling (the timetabling problem) and vertex coloring in graphs for map coloring and the assignment of frequencies in GSM mobile phone networks. Finally, we revisit the classical problem of finding re-entrant knight’s tours on a chessboard using Hamiltonian circuits in graphs.
3. INTRODUCTION Graph theory is rapidly moving into the mainstream of mathematics mainly because of its applications in diverse fields which include biochemistry (genomics), electrical engineering (communications networks and coding theory), computer science (algorithms and computations) and operations research (scheduling). The wide scope of these and other applications has been well-documented cf. [5][19]. The powerful combinatorial methods found in graph theory have also been used to prove significant and well-known results in a variety of areas in mathematics itself. The best known of these methods are related to a part of graph theory called matchings, and the results from this area are used to prove Dilworth’s chain decomposition theorem for finite partially ordered sets. An application of matching in graph theory shows that there is a common set of left and right coset representatives of a subgroup in a finite group. This result played an important role in Dharwadker’s 2000 proof of the four-color theorem [8][18]. The existence of matchingsin certain infinite bipartite graphs played an important role in Laczkovich’s affirmative answer to Tarski’s 1925 problem of whether a circle is piecewise congruent to a square. The proof of the existence of a subset of the real numbers R that is non-measurable in the Lebesgue sense is due to Thomas [21]. Surprisingly, this theorem can be proved using only discrete mathematics (bipartite graphs). There are many such examples of applications of graph theory to other parts of mathematics, but they remain scattered in the literature [3][16]. In this paper, we present a few selected applications of graph theory to other parts of mathematics and to various other fields in general.
4. TheCantor-Schröder-BernsteinTheorem(Cantor-Schröder-Bernstein). For the sets A and B, if there is an injective mapping f: A -> B and an injective mapping g: B -> A, then there is a bijection from A onto B, that is, A and B have the same cardinality Proof. Without loss of generality, assume A and B to be disjoint. Define a bipartite graph G = (A, B, E), where x,y∈ E if and only if either f(x) = y or g(y) = x, x∈A, y∈B. By the hypothesis, 1 ≤ d(v) ≤ 2 for each v of G. Therefore, each component of G is either a one-way infinite path (that is, a path of the form x0, x1, ..., xn, ...), or a two-way infinite path (of the form ..., x-n, ..., x-1 , x0, x1, ..., xn, ...), or a cycle of even length with more than two vertices, or an edge. Note that a finite path of length greater or equal to two cannot be a component of G. Thus, in each component there is a set of edges such that each vertex in the component is incident with precisely one of these edges. Hence, in each component, the subset of vertices from A is of the same cardinality as the subset of vertices from B. ☐ 1.2. Remark. Cantor inferred the result as a corollary of the well-ordering principle. The above argument shows that the result can be proved without using the axiom of choice. Figure 1.1. Types of components of the bipartite graph G
5. Fermat’s (Little) TheoremThere are many proofs of Fermat’s Little Theorem. The first known proof was communicated by Euler in his letter of March 6, 1742 to Goldbach. The idea of the graph theoretic proof given below can be found in [12] where this method, together with some number theoretic results, was used to prove Euler’s generalization to non-prime modulus Theorem (Fermat). Let a be a natural number and let p be a prime such that a is not divisible by p. Then, ap - a is divisible by p. Proof. Consider the graph G = (V, E), where the vertex set V is the set of all sequences (a1, a2, ..., ap) of natural numbers between 1 and a (inclusive), with ai ≠ aj for some i ≠ j. Clearly, V has ap - a elements. Let u = (u1, u2, ..., up), v = (up, u1, ..., up-1) ∈V. Then, we say uv ∈E. With this assumption, each vertex of G is of degree 2. So, each component of G is a cycle of length p. Therefore, the number of components is (ap - a) / p. That is, p | (ap - a). ☐ Figure 2.1. The graph G for a = 2 and p = 3
7. The SNP Assembly Problem In computational biochemistry there are many situations where we wish to resolve conflicts between sequences in a sample by excluding some of the sequences. Of course, exactly what constitutes a conflict must be precisely defined in the biochemical context. We define a conflict graph where the vertices represent the sequences in the sample and there is an edge between two vertices if and only if there is a conflict between the corresponding sequences. The aim is to remove the fewest possible sequences that will eliminate all conflicts. Recall that given a simple graph G, a vertex cover C is a subset of the vertices such that every edge has at least one end in C. Thus, the aim is to find a minimum vertex cover in the conflict graph G (in general, this is known to be a NP-complete problem [13]). We look at a specific example of the SNP assembly problem given in [15] and show how to solve this problem using the vertex cover algorithm [6]. A Single Nucleotide Polymorphism (SNP, pronounced “snip”) [15] is a single base mutation in DNA. It is known that SNPs are the most common source of genetic polymorphism in the human genome (about 90% of all human DNA polymorphisms).
9. The SNP Assembly Problem [15] is defined as follows. A SNP assembly is a triple (S, F, R) where S = {s1, ..., sn} is a set of n SNPs, F = {f1, ..., fm} is a set of m fragments and R is a relation R: S×F -> {0, A, B} indicating whether a SNP si ∈S does not occur on a fragment fj ∈F (marked by 0) or if occurring, the non-zero value of si (A or B). Two SNPs si and sj are defined to be in conflict when there exist two fragments fk and fl such that exactly three of R(si, fk), R(si, fl), R(sj, fk), R(sj, fl) have the same non-zero value and exactly one has the opposing non-zero value. The problem is to remove the fewest possible SNPs that will eliminate all conflicts. The following example from [15] is shown in the table below. Note that the relation R is only defined for a subset of S×F obtained from experimental values.
11. Note, for instance, that s1 and s5 are in conflict because R(s1, f2) = B, R(s1, f5) = B, R(s5, f2) = B, R(s5, f5) = A. Again, s4 and s6 are in conflict because R(s4, f1) = A, R(s4, f3) = A, R(s6, f1) = B, R(s6, f3) = A. Similarly, all pairs of conflicting SNPs are easily determind from the table. The conflict graph G corresponding to this SNP assembly problem is shown below in figure 4.3.
12. We now use the vertex cover algorithm [6] to find minimal vertex covers in the conflict graph G. The input is the number of vertices 6, followed by the adjacency matrix of G shown below in figure 4.4. The entry in row i and column j of the adjacency matrix is 1 if the vertices si and sj have an edge in the conflict graph and 0 otherwise.
13. The vertex cover program [6] finds two distinct minimum vertex covers, shown in figure 4.5. Minimum vertex cover s1,s4
15. computer network security A team of computer scientists led by Eric Filiol[11] at the Virology and Cryptology Lab, ESAT, and the French Navy, ESCANSIC, have recently used the vertex cover algorithm [6] 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.
16.
17. The set {2, 4, 5} is a minimum vertex cover in this computer network The simulation was carried out on a large internet-like virtual 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 5.1 above shows a simple computer network and a corresponding minimum vertex cover {2, 4, 5}.
18. 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 [8][18][1]. 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.
19.
20.
21. Above figure is The dual graph of the map of India We use the vertex coloring algorithm [7] to find a proper coloring of the map of India with four colors, see figures 7.1 and 7.2 above. The GroupeSpécial 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 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 cells in the immediate vicinity. GSM networks operate in only 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 7.3 below.
22.
23. The Timetabling Problem In a college there are m professors x1, x2, …, xm and n subjects y1, y2, …, yn to be taught. Given that professor xi is required (and able) to teach subject yj for pij periods (p = [ pij] is called the teaching requirement matrix), the college administration wishes to make a timetable using the minimum possible number of periods. This is known as the timetabling problem [4] and can be solved using the following strategy. Construct a bipartite multigraphG with vertices x1, x2, …, xm, y1, y2, …, yn such that vertices xi and yj are connected by pij edges. We presume that in any one period each professor can teach at most one subject and that each subject can be taught by at most one professor. Consider, first, a single period. The timetable for this single period corresponds to a matching in the graph and, conversely, each matching corresponds to a possible assignment of professors to subjects taught during this period. Thus, the solution to the timetabling problem consists of partitioning the edges of G into the minimum number of matchings. Equivalently, we must properly color the edges of G with the minimum number of colors. We shall show yet another way of solving the problem using the vertex coloring algorithm [7]. Recall that the line graph L(G) of G has as vertices the edges of G and two vertices in L(G) are connected by an edge if and only if the corresponding edges in G have a vertex in common. The line graph L(G) is a simple graph and a proper vertex coloring of L(G) yields a proper edge coloring of G using the same number of colors. Thus, to solve the timetabling problem, it suffices to find a minimum proper vertex coloring of L(G) using [7]. We demonstrate the solution with a small example. Suppose there are four professors x1, x2, x3, x4 and five subjects y1, y2, y3, y4, y5 to be taught [4]. The teaching requirement matrix p = [ pij] is given below in figure 6.1.
25. We first construct the bipartite multigraphG shown below in figure below
26. Next, we construct the line graph L(G). The adjacency matrix of L(G) is given below
27. Above figure is The adjacency matrix of L(G)Now, we use the vertex coloring algorithm [7] to find a minimum proper 4-coloring of the vertices of L(G). A minimum proper 4-coloring of the vertices of L(G)
28. Vertex Coloring: (1, green), (2, red), (3, blue), (4, yellow), (5, yellow), (6, green), (7, green), (8, yellow), (9, red), (10, blue), (11, yellow). This, in turn, yields a minimum proper edge 4-coloring of the bipartite multigraphG:
29. Edge Coloring: ({x1, y1}, green), ({x1, y1}, red), ({x1, y3}, blue), ({x1, y4}, yellow), ({x2, y2}, yellow), ({x2, y4}, green), ({x3, y2}, green), ({x3, y3}, yellow), ({x3, y4}, red), ({x4, y4}, blue), ({x4, y5}, yellow). Interpret the colors green, red, blue, yellow as periods 1, 2, 3, 4 respectively. Then, from the edge coloring of G, we obtain a solution of the given timetabling problem as shown below in figure 6.5. The timetable
30. References [1] K. Appel and W. Haken, Every Planar Map is Four Colorable, Bull. Amer. Math. Soc. 82 (1976) 711-712. [2] L. Babai, Some applications of graph contractions, J. Graph Theory, Vol. 1 (1977) 125-130. [3] E. Bertram and P. Horak, Some applications of graph theory to other parts of mathematics, The Mathematical Intelligencer (Springer-Verlag, New York) (1999) 6-11. [4] J.A. Bondy and U.S.R. Murty, Graph Theory with Applications, 1976, Elsevier Science Publishing Company Inc. [5] L. Caccetta and K. Vijayan, Applications of graph theory, Fourteenth Australasian Conference on Combinatorial Mathematics and Computing (Dunedin, 1986), Ars. Combin., Vol. 23 (1987) 21-77. [6] AshayDharwadker, The Vertex Cover Algorithm, 2006, [7] AshayDharwadker, The Vertex Coloring Algorithm, 2006, [8] AshayDharwadker, A New Proof of The Four Colour Theorem, 2000, [9] AshayDharwadker, A New Algorithm for finding Hamiltonian Circuits, 2004, [10] L. Euler, Solution d'une question curieuse qui ne paroitsoumise a aucuneanalyse, Mémoires de l'Académie Royale des Sciences et Belles Lettres de Berlin, Année 1759 15, 310-337, 1766. [11] Eric Filiol, Edouard Franc, Alessandro Gubbioli, Benoit Moquet and Guillaume Roblot, Combinatorial Optimisation of Worm Propagation on an Unknown Network, Proc. World Acad. Science, Engineering and Technology, Vol 23, August 2007, [12] K. Heinrich and P. Horak, Euler’s theorem, Am. Math. Monthly, Vol. 101 (1994) 260. [13] R.M. Karp, Reducibility among combinatorial problems, Complexity of Computer Computations, Plenum Press, 1972. [14] D. König, Theoriederendlichen und unendlichengraphen, AkademischeVerlagsgesllschaft, Leipzing (1936), reprinted by Chelsea, New York (1950). [15] G. Lancia, V. Afna, S. Istrail, L. Lippert, and R. Schwartz, SNPs Problems, Complexity and Algorithms, ESA 2002, LNCS 2161, pp. 182-193, 2001. Springer-Verlag 2001. [16] L. Lovasz, L. Pyber, D. J. A. Welsh and G. M. Ziegler, Combinatorics in pure mathematics, in Handbook of Combinatorics, Elsevier Sciences B. V., Amsterdam (1996). [17] H. J. R. Murray, A History of Chess, Oxford University Press, 1913. [18] ShariefuddinPirzada and AshayDharwadker, Graph Theory, Orient Longman and Universities Press of India, 2007. [19] F. S. Roberts, Graph theory and its applications to the problems of society, CBMS-NSF Monograph 29, SIAM Publications, Philadelphia (1978). [20] J. P. Serre, GroupesDiscretes, Extrait de I’Annuaire du College de France, Paris (1970). [21] R. Thomas, A combinatorial construction of a non-measurable set, Am. Math. Monthly 92 (1985) 421-422.
31. THANKS FROM MILAN JOSHI AKOLA MO:- 9890444163 Email:- mlnjsh@gmail.com
32. SPECIAL THANKS TO ShariefuddinPirzada Department of MathematicsUniversity of KashmirSrinagarJammu and Kashmir 190006India AshayDharwadker Institute of MathematicsH-501 PalamViharDistrict GurgaonHaryana 122017India