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.
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 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.
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.
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.
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.
This document provides an introduction to fundamental concepts in graph theory. It defines what a graph is composed of and different graph types including simple graphs, directed graphs, bipartite graphs, and complete graphs. It discusses graph terminology such as vertices, edges, paths, cycles, components, and subgraphs. It also covers graph properties like connectivity, degrees, isomorphism, and graph coloring. Examples are provided to illustrate key graph concepts and theorems are stated about properties of graphs like the Petersen graph and graph components.
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 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.
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 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.
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.
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.
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.
This document provides an introduction to fundamental concepts in graph theory. It defines what a graph is composed of and different graph types including simple graphs, directed graphs, bipartite graphs, and complete graphs. It discusses graph terminology such as vertices, edges, paths, cycles, components, and subgraphs. It also covers graph properties like connectivity, degrees, isomorphism, and graph coloring. Examples are provided to illustrate key graph concepts and theorems are stated about properties of graphs like the Petersen graph and graph components.
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 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 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.
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.
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 some basic concepts in graph theory, including:
- A graph G is defined as a pair (V,E) where V is the set of vertices and E is the set of edges.
- Edges connect pairs of vertices and can be directed or undirected. Special types of edges include parallel edges and loops.
- Special graphs include simple graphs without parallel edges/loops, weighted graphs with numerical edge weights, and complete graphs where all vertex pairs are connected.
- Graphs can be represented by adjacency matrices and incidence matrices showing vertex-edge connections.
- Paths and cycles traverse vertices and edges, with Euler cycles passing through every edge once.
The document discusses different matrix representations of graphs:
1) Incidence matrix shows which edges are incident to each vertex with 1s and 0s.
2) Adjacency matrix shows which vertices are adjacent to each other with 1s and 0s.
3) Cut-set matrix shows which edges are part of given cut sets that disconnect the graph with 1s and 0s.
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.
The document defines key concepts in graph theory including:
- Types of graphs such as simple graphs, connected graphs, and regular graphs.
- Graph terminology like vertices, edges, walks, paths, and subgraphs.
- Special graphs like Hamiltonian and Euler graphs.
- Graph coloring problems including vertex coloring and edge coloring.
- Examples are given to illustrate graph concepts and properties.
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.
Graph coloring involves assigning colors to the vertices of a graph such that no two adjacent vertices have the same color. This is called a proper coloring. The minimum number of colors needed for a proper coloring is called the chromatic number. Chromatic partitioning and the chromatic polynomial are other concepts related to graph coloring. Graph coloring has applications in Sudoku puzzles, traffic signal design, scheduling exams, and other areas where resources need to be allocated without conflicts.
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 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.
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.
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.
The document discusses graph theory and provides definitions and examples of various graph concepts. It defines what a graph is consisting of vertices and edges. It also defines different types of graphs such as simple graphs, multigraphs, digraphs and provides examples. It discusses graph terminology, models, degree of graphs, handshaking lemma, special graphs and applications. It also provides explanations of planar graphs, Euler's formula and graph coloring.
The document provides definitions and examples related to graphs. It defines graphs, directed graphs, and terminology like vertices, edges, degrees, adjacency, and isomorphism. It discusses representations of graphs using adjacency lists and matrices. It also covers special types of graphs like trees, cycles, and bipartite graphs. Key concepts are illustrated with examples.
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.
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.
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 some basic concepts in graph theory, including:
- A graph G is defined as a pair (V,E) where V is the set of vertices and E is the set of edges.
- Edges connect pairs of vertices and can be directed or undirected. Special types of edges include parallel edges and loops.
- Special graphs include simple graphs without parallel edges/loops, weighted graphs with numerical edge weights, and complete graphs where all vertex pairs are connected.
- Graphs can be represented by adjacency matrices and incidence matrices showing vertex-edge connections.
- Paths and cycles traverse vertices and edges, with Euler cycles passing through every edge once.
The document discusses different matrix representations of graphs:
1) Incidence matrix shows which edges are incident to each vertex with 1s and 0s.
2) Adjacency matrix shows which vertices are adjacent to each other with 1s and 0s.
3) Cut-set matrix shows which edges are part of given cut sets that disconnect the graph with 1s and 0s.
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.
The document defines key concepts in graph theory including:
- Types of graphs such as simple graphs, connected graphs, and regular graphs.
- Graph terminology like vertices, edges, walks, paths, and subgraphs.
- Special graphs like Hamiltonian and Euler graphs.
- Graph coloring problems including vertex coloring and edge coloring.
- Examples are given to illustrate graph concepts and properties.
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.
Graph coloring involves assigning colors to the vertices of a graph such that no two adjacent vertices have the same color. This is called a proper coloring. The minimum number of colors needed for a proper coloring is called the chromatic number. Chromatic partitioning and the chromatic polynomial are other concepts related to graph coloring. Graph coloring has applications in Sudoku puzzles, traffic signal design, scheduling exams, and other areas where resources need to be allocated without conflicts.
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 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.
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.
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.
The document discusses graph theory and provides definitions and examples of various graph concepts. It defines what a graph is consisting of vertices and edges. It also defines different types of graphs such as simple graphs, multigraphs, digraphs and provides examples. It discusses graph terminology, models, degree of graphs, handshaking lemma, special graphs and applications. It also provides explanations of planar graphs, Euler's formula and graph coloring.
The document provides definitions and examples related to graphs. It defines graphs, directed graphs, and terminology like vertices, edges, degrees, adjacency, and isomorphism. It discusses representations of graphs using adjacency lists and matrices. It also covers special types of graphs like trees, cycles, and bipartite graphs. Key concepts are illustrated with examples.
This document provides an introduction to graph theory concepts. It defines what a graph is consisting of vertices and edges. It discusses different types of graphs like simple graphs, multigraphs, digraphs and their properties. It introduces concepts like degrees of vertices, handshaking lemma, planar graphs, Euler's formula, bipartite graphs and graph coloring. It provides examples of special graphs like complete graphs, cycles, wheels and hypercubes. It discusses applications of graphs in areas like job assignments and local area networks. The document also summarizes theorems regarding planar graphs like Kuratowski's theorem stating conditions for a graph to be non-planar.
This document provides an overview of graph theory. It defines various graph types including simple graphs, multigraphs, pseudographs, directed graphs, and labeled graphs. It also defines key graph terminology such as vertices, edges, degree, adjacency, connectivity, and planar graphs. Graph theory has many applications in fields like transportation, computer networks, and chemistry for modeling relationships between objects.
This document provides an overview of graph theory concepts. It defines what a graph is consisting of vertices and edges. It describes different types of graphs such as directed vs undirected, simple vs complex graphs. It introduces common graph terminology like degree of a vertex, adjacent/incident vertices, and connectivity. Examples of applications are given such as transportation networks, web graphs, and scheduling problems. Special graph cases like complete graphs and cycles are also defined.
Graph theory is the study of graphs, which are mathematical structures used to model pairwise relations between objects. A graph consists of vertices and edges connecting pairs of vertices. There are many types of graphs including trees, which are connected acyclic graphs. Spanning trees are subgraphs of a graph that connect all vertices using the minimum number of edges. Key concepts in graph theory include paths, connectedness, cycles, and isomorphism between graphs.
The document discusses various graph theory concepts including:
- Types of graphs such as simple graphs, multigraphs, pseudographs, directed graphs, and directed multigraphs which differ based on allowed edge connections.
- Graph terminology including vertices, edges, degrees, adjacency, incidence, paths, cycles, and representations using adjacency lists and matrices.
- Weighted graphs and algorithms for finding shortest paths such as Dijkstra's algorithm.
- Euler and Hamilton paths/circuits and conditions for their existence.
- The traveling salesman problem of finding the shortest circuit visiting all vertices.
1) Graph theory concepts such as undirected graphs, directed graphs, weighted graphs, mixed graphs, simple graphs, multigraphs, and pseudographs are introduced. Common graph terminology like vertices, edges, loops, degrees, adjacency and connectivity are defined.
2) Different types of graphs are discussed including trees, forests, cycles, wheels, complete graphs, bipartite graphs, and n-cubes. Matrix representations using adjacency matrices are also covered.
3) The document touches on graph isomorphism, Eulerian graphs, Hamiltonian graphs, and algorithms for determining graph properties like connectivity and number of components.
This document defines and provides examples of different types of graphs, including finite and infinite graphs, simple graphs, complete graphs, bipartite graphs, and regular graphs. It introduces key graph terminology like vertices, edges, degrees, adjacency, and isolation. Examples are provided to illustrate concepts like the handshake theorem, determining if certain degree sequences can form graphs, and drawing regular graphs.
Graph terminologies & special type graphsNabeel Ahsen
The document discusses various graph terminologies and special types of graphs. It defines undirected and directed graphs, and describes degrees, handshaking theorem, and other properties. Special graph types covered include complete graphs, cycles, wheels, n-cubes, and bipartite graphs. It provides examples of constructing new graphs from existing ones and an application using a bipartite graph model for employee skills and job assignments.
This document provides an introduction to graph theory, including basic terminology and concepts. It defines what a graph is mathematically as a collection of vertices and edges. It describes different types of graphs such as simple graphs, multiple graphs, weighted graphs, finite and infinite graphs, labeled graphs, and directed graphs. It also defines graph theory terms like adjacency, incidence, degree of a vertex, isomorphism, subgraphs, and graph operations like union and intersection. The document provides examples to illustrate each term and concept.
This document provides an introduction to graph theory, including basic terminology and concepts. It defines what a graph is mathematically as a collection of vertices and edges. It describes different types of graphs such as simple graphs, multiple graphs, weighted graphs, finite and infinite graphs, labeled graphs, and directed graphs. It also defines graph theory terms like adjacency, incidence, degree of a vertex, isomorphism, subgraphs, and graph operations like union and intersection. The document provides examples to illustrate each term and concept.
1. Graph and Graph Terminologiesimp.pptxswapnilbs2728
There are five main categories of graphs: simple graphs, multigraphs, pseudographs, directed graphs, and directed multigraphs. An undirected graph G consists of a set of vertices V and a set of edges E that connect the vertices. A directed graph consists of vertices V and directed edges E that have an initial and terminal vertex. There are several special types of simple graphs including complete graphs, cycles, wheels, and bipartite graphs.
Graph algorithms can be used to model and solve many real world problems. Some famous graph problems include the traveling salesman problem of planning the shortest route to visit all cities once, and the four color theorem about coloring maps with four colors so no adjacent regions have the same color. Graphs are mathematical structures used to represent pairwise relationships between objects. They are made up of vertices connected by edges, and common graph algorithms involve finding shortest paths, matching items, or modeling flows through networks.
This document provides an introduction to graph theory. It defines key graph terminology like vertices, edges, directed and undirected graphs, paths, and connectivity. Examples are given to illustrate different graph types like trees, cycles, and complete graphs. Common graph representations like adjacency matrices and lists are also described. The document outlines theorems and properties of graphs, and discusses applications of graph theory concepts in areas like computer networks and chemistry.
This document contains definitions and explanations of various graph theory terms in the form of 26 multiple choice questions. Some key terms defined and explained include: graph, simple graph, connected graph, components, tree, rooted tree, binary tree, walk, path, circuit, degree, adjacency, incidence, isomorphism, subgraph, Euler graph, and Hamiltonian path. Examples are provided to illustrate many of the graph theory concepts discussed.
This document contains 26 multiple choice questions and their answers related to graph theory. It begins by defining key graph theory terms like graphs, vertices, edges, simple graphs, and applications of graph theory. It then discusses incidence, adjacency, degrees, finite and infinite graphs, isolated and pendant vertices, null graphs, and multigraphs. The document also defines complete graphs, regular graphs, cycles, isomorphism, subgraphs, walks, paths, circuits, connectivity, components, Euler graphs, Hamiltonian circuits/paths, trees, properties of trees, distance in trees, eccentricity, center, distance metric, radius, diameter, rooted trees, and binary rooted trees.
The document discusses graphs and graph theory. It defines graphs as non-linear data structures used to model networks and relationships. The key types of graphs are undirected graphs, where edges have no orientation, and directed graphs, where edges have orientation. Graph traversal algorithms like depth-first search and breadth-first search are discussed. Common graph terminology is defined, including vertices, edges, paths, cycles, degrees. Different graph representations like adjacency matrices and adjacency lists are also covered. Applications of graphs include modeling networks, routes, and relationships.
Graphs are propular to visualize a problem . Matrix representation is use to convert the graph in a form that used by the computer . This will help to get the efficent solution also provide a lots of mathematical equation .
A graph G = (V, E) consists of a set of vertices V = { V1, V2, . . . } and set of edges E = { E1, E2, . . . }. The set of unordered pairs of distinct vertices whose elements are called edges of graph G such that each edge is identified with an unordered pair (Vi, Vj) of vertices.The vertices (Vi, Vj) are said to be adjacent if there is an edge Ek which is associated to Vi and Vj. In such a case Vi and Vj are called end points and the edge Ek is said to be connect/joint of Vi and Vj.
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.
ISO/IEC 27001, ISO/IEC 42001, and GDPR: Best Practices for Implementation and...PECB
Denis is a dynamic and results-driven Chief Information Officer (CIO) with a distinguished career spanning information systems analysis and technical project management. With a proven track record of spearheading the design and delivery of cutting-edge Information Management solutions, he has consistently elevated business operations, streamlined reporting functions, and maximized process efficiency.
Certified as an ISO/IEC 27001: Information Security Management Systems (ISMS) Lead Implementer, Data Protection Officer, and Cyber Risks Analyst, Denis brings a heightened focus on data security, privacy, and cyber resilience to every endeavor.
His expertise extends across a diverse spectrum of reporting, database, and web development applications, underpinned by an exceptional grasp of data storage and virtualization technologies. His proficiency in application testing, database administration, and data cleansing ensures seamless execution of complex projects.
What sets Denis apart is his comprehensive understanding of Business and Systems Analysis technologies, honed through involvement in all phases of the Software Development Lifecycle (SDLC). From meticulous requirements gathering to precise analysis, innovative design, rigorous development, thorough testing, and successful implementation, he has consistently delivered exceptional results.
Throughout his career, he has taken on multifaceted roles, from leading technical project management teams to owning solutions that drive operational excellence. His conscientious and proactive approach is unwavering, whether he is working independently or collaboratively within a team. His ability to connect with colleagues on a personal level underscores his commitment to fostering a harmonious and productive workplace environment.
Date: May 29, 2024
Tags: Information Security, ISO/IEC 27001, ISO/IEC 42001, Artificial Intelligence, GDPR
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How to Fix the Import Error in the Odoo 17Celine George
An import error occurs when a program fails to import a module or library, disrupting its execution. In languages like Python, this issue arises when the specified module cannot be found or accessed, hindering the program's functionality. Resolving import errors is crucial for maintaining smooth software operation and uninterrupted development processes.
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.
This presentation includes basic of PCOS their pathology and treatment and also Ayurveda correlation of PCOS and Ayurvedic line of treatment mentioned in classics.
This slide is special for master students (MIBS & MIFB) in UUM. Also useful for readers who are interested in the topic of contemporary Islamic banking.
This presentation was provided by Steph Pollock of The American Psychological Association’s Journals Program, and Damita Snow, of The American Society of Civil Engineers (ASCE), for the initial session of NISO's 2024 Training Series "DEIA in the Scholarly Landscape." Session One: 'Setting Expectations: a DEIA Primer,' was held June 6, 2024.
it describes the bony anatomy including the femoral head , acetabulum, labrum . also discusses the capsule , ligaments . muscle that act on the hip joint and the range of motion are outlined. factors affecting hip joint stability and weight transmission through the joint are summarized.
How to Add Chatter in the odoo 17 ERP ModuleCeline George
In Odoo, the chatter is like a chat tool that helps you work together on records. You can leave notes and track things, making it easier to talk with your team and partners. Inside chatter, all communication history, activity, and changes will be displayed.
2. Introduction
Graph
What is a graph G?
It is a pair G = (V, E),
where
V = V(G) = set of vertices
E = E(G) = set of edges
An elemnet of a set E is
generally
denoted as e=(u,v) or e=(v,u),
wherev u,v,ЄV.
Example:
V = {s, u, v, w, x, y, z}
E = {(x,s), (x,v), (x,u), (v,w), (s,v), (s,u), (s,w), (s,y), (w,y), (u,y),
(u,z),(y,z)}
3. Basic Terminology:
1)Adjacency:
-Two vertices v1 and v2 in a graph G are said to be
adjacent to each other iff they are end vertices of the
same edge e.
2) Incidence:
-If the vertex u is an end vertex of the edge e then the
edge e is said to be incident on vertex u .
Here, adjacent pairs of vertices:
{v2,v3},{v3,v4},{v2,v4} and
edge e1 is incident on vertices
v1 and v2.
4. Special edges
Parallel edges:
If Two or more edges have
same terminal vertices
then these edges are called as
parallel edges.
In the example, a and b are
joined by two parallel edges
Loop(Self loop):
If both the end vertices of an
edges are same then the edge is
called a loop.
In the example, vertex d has a
loop,e8 is loop.
5. Degree of a vertex
The degree of a vertex v, denoted by d(v), is the
number of edges incident on v
From example,
Degree of each vertex is
d(a)=4
d(b)=3
d(c)=4
d(d)=6
d(e)=4
d(f)=4
d(g)=3
Note:
If there is self loop then in the
definition of degree of vertex it is
counted twice
6. Pendant Vertex:
A vertex with degree one is called pendant vertex.
i.e deg (v) =1
Isolated Vertex:
A vertex with degree zero is called Isolated vertex.
deg (v) = 0
Representation Example:
For V = {u, v, w} ,
E = { {u, w}, (u, v) }},
deg (u) = 2, deg (v) = 1, deg
(w) = 1, deg (k) = 0,
w and v are pendant ,
k is isolated
7. Degree of the graph:
Sum of the degrees of all vertices in graph G is
called degree of the graph G.
Notation:
d(G)=degree of graph G.
d(G)= ∑ d(vi)
viЄG
i.e
d(v1)=5,d(v2)=5,d(v3)=3,
d(v4)=5,d(v5)=1,d(v6)=1
d(G)=20
8. EX: How many edges are there in a graph with
10 vertices each of degree 6?
Solution:
In graph of 10 vertices with each of degree 6.
d(G)=10X6=60
Each edge gives two degree.
Threfore,
Total number of edges in such graph=60/2=30
Therefore,G has 30 edges.
9. Hand shaking lemma:
Statement:
The sum of the degrees of all vertices
in any graph is always even number.
i.e ∑ d(v)=2.e,
vЄG
where, e= Total number of edges.
10. EX: Does there exists a party of 11
professors such that each one has exactly 7
friends in themselves?
Solution:
If we translate given situation of party in graph by
taking each professor as a vertex and thire
friendship by edges.
Then we get a graph G on 11 vertices with degree
of each vertex is 7.
i,e d(G)= 11X7=77 which is odd.
A contradiction to handshaking lemma
There does not exists such party.
11. EX: how many edges are there in the group with
8 vertices each of degree 5?
Solution : There are 8 vertices of degree 5 in a given graph
G.
Total degree of such 8 vertices
∑ d(G)= 8X5=40,
vЄG
By handshaking lemma,
∑ d(G) =2.e,
vЄG
40=2e
e=20
There are 20 edges int he graph.
12. Special graphs
• Simple graph
A graph without loops or parallel edges.
• Null Graph:
A graph with n vertices without edges.
Notation:
Nn:Null graph with n vertices.
• For e.g: . . .
N1 N2
16. Types of Graph.
Multigraph:
• A graph with parallel edges
but not loop is called Multi
graph.
Compound graph
(Pseudograph):
• A graph which contains
loops or parallel edges.
18. Complete Graph:
• Let n > 3
• A simple graph G
is called a complete graph
• if there is an edge between
every pair of vertices
• Notation:
• Kn = complete graph on n vertices
The figure represents K5
19. Properties of complete graph of n
vertices:
• Each vertex has degree (n-1).
• Number of edges=
𝑛(𝑛−1)
2
21. EX:Show that there does not exists a simple
graph with 8 vertices and 29 edges.
Solution:
Let G be a simple graph with 8 vertices.
Maximum number of edges with n vertices is
n(n-1)/2
Therefore,G has at most 8X7/2=28 edges.
Therefore,There does not exists a simple graph
with 8 vertices and 29 edges.
22. EX: Determine minimum number of vertices in a simple
graph with 30 edges.
Solution:
Let G be a simple graph with n vertices.
Maximum number of edges with n vertices are
𝑛(𝑛 − 1)
2
but G has exactly 30 edges
𝑛(𝑛−1)
2
≥30 for minimum value of n.
n(n-1)-60≥0
=>9.8-60≥0
Therefore n=9 is the minimum number of vertices for
simplegraph with 30 edges.
23. EX:Find the smallest integer n such that kn has at least
600 edges.
Solution:
Complete graph on n vertices has
𝑛(𝑛−1)
2
edges.
but the graph requires 600 edges.
𝑛(𝑛−1)
2
≥600
n(n-1)≥1200
36X35≥1200
n=36
Therefore, No of vertices of such required graph n=36
24. Regular Graph:
If all vertices of a graph G have same
degree then, G is called as a Regular
graph.
Remark: If every vertex has degree m
then G is said to be regular graph of
degree m or m- regular graph.
Notation:
mRn=-m Regular graph on n vertices
27. Properties of Regular graph of n
vertices:
1) Each vertex is of same degree m
2) Number of edges= m*n/2.
3) Degree of graph d(G)= m*n.
4) Every complete graph is regular graph but
converse is not true.
28. EX: Does every regular graph
complete? Justify
Solution: Every regular graph is not complete.
Justification: We have
2R4-2regular graph with 4
vertices.But not complete graph
29. EX: Draw a 3-regular graph on 6 vertices?
EX: Draw all possible 2-regular graphs on 4 vertices.
• Solution:3-regular graph
30. Bipartite Graph
• A bipartite graph G is if the vertex set V can be
partitioned into two non empty disjoint
subsets V(G1) and V(G2) i.e such graph is
that
31. V(G1) V(G2)
• V(G) = V(G1)U V(G2),
|V( G1)| = m, |V(G2)| =n
V(G1) ∩V(G2) = φ
No edges exist between any two vertices in the same
subset
V(Gk), k =1,2
33. EX: Draw the bipartite graph which is
not regular graph.
34. Complete bipartite graph Km,n
• A bipartite graph is the
complete bipartite graph
Km,n
• if every vertex in V(G1) is
joined to a vertex in V(G2)
and conversely,
• |V(G1)| = m |V(G2)| = n
• then this bipartite graph
• complete bipartite graph
Km,n
35. Properties of complete bipertite graph
with vertex set V1 contains m vertices
and vertex set V2 contains n vertices
1) Each vertex of vertex set V1 has degree n and
each vertex of vertex set V2 has degree m.
2) Number of edges are mXn.
3) Every complete bipartite graph is bipartite but
converse is not true.i.e every bipartite graph may
not be complete bipartite.
4) d(G)=2*m*n
36. EX: Find number of edges is K5,8 graph.
Solution:
K5,8 graph is complete bipartite graph in which firest
vertex set V1 contains 5 vertices and second V2
contains 8 vertices.
Number of edges =5X8=40
EX: Find number of edges is K7,8 graph.
37. EX: Draw the following graphs
i)Non complete bipartite graph
ii) Complete graph which is complete bipartite graph.
iii)Regular graph but not complete.
iv) 3R6 i.e 3regular graph with 6 vertices.
• Solution:
Non complete bipartite
graph
38. • Complete graph which
is complete bipartite
graph
• 3R6 i.e 3regular graph
with 6 vertices.
• Regular graph but not
complete
39. EX: Daw the following graphs
i) Non complete bipartite graph
ii) Complete graph which is complete bipartite
• Solution:
• i) Non complete
bipartite graph
• Complete graph which
is complete bipartite
44. Directed graphs (Digraphs)
• G is a directed graph or
digraph if each edge has
been associated with an
ordered pair of vertices,
i.e. each edge has a
direction.
45. Terminology – Directed graphs
For the edge (u, v), u is adjacent to v OR v is
adjacent from u,
u – Initial vertex, v – Terminal vertex
In-degree (deg- (u)): The number of edges
incident into the vertex u.
Out-degree (deg+ (u)): The number of edges
incident out of a vertex u.
46. Arcs:
Directeg edges are called Arcs.
Initial Vertex:
Starting vertex of arc is called itial vertex.
Terminal Vertex:
Ending vertex of arc is called terminal vertex.
Multiple arcs:
If two vertices are joined by more than one arcs
with the same direction ,such arc are called
multiple arcs.
47. • V={V1,V2,V3}
• A={(v2,v1),(v2,v3),(v3,v2),
(v3v1)}
• e1=(v3,v1)
• e2=(v2,v1),e3=(v2,v1)
• e4=(v3,v2),e5=(v2,v3)
• Arcs e2 and e3 are
multiple arcs,but e4 ,e5
are not multiple arcs
because tire direction are
different.
49. Remark:
1. Each arc gives one indegree to terminal vertex and
outdegree to initial vertex.
2. Sum of all indegrees of the digraph eqauls to sum
of all outdegress.
∑d-(u)=∑d+(u)
uЄv uЄv
Pendant vertex:
In a digraph a vertex having d+(u)+d-(u)=1 is called
pendant vertex.
50. Find Indegree and Outdegree of following
Digraph.
Solution:
Indegrees of v1,v2,v3 are
d-(v1)=3
d(v2)=1
d-(v3)=1
Outdegrees of v1,v2,v3 are
d+(v1)=0
d+(v2)=3
d+(v3)=2
V3 is pendant vertex.
51. Types of diagraph:
Simple Digraph:
A digraph without self loop or multiple arcs is called
simple digraph.
54. Symmetric Digraph:
A digraph D is called symmetric if for every directed
edge (a,b) in D ,then there must be directed edge (b,a)
in D is called as symmetric digraph.
• For.e.g
56. Asymmetric digraph:
A digraph D in which there exists at the most one
directed edge between evry pair of vertices with may
or may not self-loop is called asymmetric digraph.
58. Balanced Digraph:
Digraph D is balanced digraph if for every vertex ,the number
of indegrees equals to number of outdegress d+(v)=d-(v) for
all vertex is called Balanced digraph.
Regular Digraph:
A balanced digraph in which
outdegree and indegree of each
vertex is same then such digraph
is called regular digraph.
d-(v1)=1 d+(v1)=1
d+(v2)=1
d(v2)=1
60. Representations of graphs
Adjacency matrix:
Definition:
If G is a graph on n vertices say v1,v2…vn then
adjacency matrix of G is the nXn matrix
A(G)=[aij]nxn
Where,
aij= number of edges between vi and vj.
=1 for self loop
61. -Rows and columns are labeled with
ordered vertices
-write a 1 if there is an edge between the
row vertex and the column vertex
and
0 if no edge exists between them
63. EX: Find the adjacency matrix of the following graph
• Solution:
v1 v2 v3 v4
v3
0 2 1 1
2 0 2 1
1 2 1 0
1 1 0 1
64. EX: Find the adjacency matrix of the
following graph
65. Incidence matrix:
Let G be a graph with n vertices v1,v2,v3...,vn and
m edges e1,e2,..,em then the incidencematrix G is
denoted by I(G) as,
I(G)=[aij]nXm where,
aij=
0 𝑖𝑓 𝑣𝑖 𝑖𝑠 𝑛𝑜𝑡 𝑖𝑛𝑐𝑖𝑑𝑒𝑛𝑡 𝑤𝑖𝑡ℎ 𝑒𝑗
1 𝑖𝑓 𝑣𝑗 𝑖𝑠 𝑖𝑛𝑐𝑖𝑑𝑒𝑛𝑡 𝑤𝑖𝑡ℎ 𝑒𝑗
2 𝑖𝑓 𝑒𝑗 𝑖𝑠 𝑎 𝑙𝑜𝑜𝑝 𝑎𝑡 𝑣𝑖
67. Observations:
• Each column contains exactly two 1’s.
• The sum of the entries in a row indicates the
degree of the corresponding vertex.
• If two columns are identical,then the
corrosponding edges are parrellel edges.
• If a row contains only zero’s then the
corrosponding vertex must be an isolated vertex.
• If a row contains a sing1,s the corresponding
vertex must be pendant vertex.
• If an edge is a loop there will be a single ‘2’ in the
column remaining entries being zero’s.
72. Isomorphism of graphs:
Definition:
Let G1(V1,E1) and G2(V2,E2) be two graphs.The
graphs G1 and G2 are said to be isomorphic if there is
bijective functions fV:V1->V2 and fE:E1->E2 such that
if u and v are end vertices of some edge e ЄE1 then
fv(u),fv(v) are end vertices of fE(e).
EX: Draw all posible non isomorphic simple graphs
with 4 vertices?
Solution:
Graph of zero edges:
73. Graph of 1 edges:
Graph of 2 edges:
Graph of 3 edges:
Graph of 4 edges:
74. Graph of 5 edges:
Graph of 6 edges:
Graph of 7 edges:
76. Number of vertices in G1 =8=number of vertices in
G2
Number of Edges in G1 =10=number of Edges in G2
In graph G1
Vertex Degree Degree of adjacent
vertices
a 3 3,2,2
b 2 3,3
c 3 2,3,2
d 2 3,3
e 3 3,2,2
f 2 3,3
g 3 3,2,2
h 2 3,3
77. In graph G2
Vertex Degree Degree of
adjacent vertices
p 3 3,3,2
q 3 2,3,3
r 2 3,2
s 2 2,3
t 3 2,3
u 2 3,2
v 2 3,2
w 3 2,3,2
78. From above table we say that in G1 vertex with
degree 3 having degree of adjacent vertex are
3,2,2 does not exist in G2.
Therefor,G1 is not isomorphic to G2.
80. Number of vertices in G1 =8=number of vertices in
G2
Number of Edges in G1 =10=number of Edges in G2
In graph G1
Vert
ex
Degree Degree of
adjacent
vertices
v1 4 3,2,2,3
v2 3 4,2,2
v3 2 4,3
v4 2 3,3
v5 2 4,3
v6 3 4,2,2
Vert
ex
Degre
e
Degree of
adjacent
vertices
u1 4 2,2,3,3
u2 2 4,3
u3 2 4,3
u4 3 4,2,2
u5 3 4,2,2
u6 2 3,3
81. From above table a bijection between
vrtices of G1 and G2 is
f(v1)=u1
f(v2)=u4
f(v3)=u3
f(v4)=u6
f(v5)=u2
f(v6)=u5
Therefore,G1 and G2 are isomorphic.