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The document discusses submatrices of circuit matrices and their properties. Some key points: - A subgraph's incidence matrix is a submatrix of the original graph's incidence matrix. - An (n-1)x(n-1) submatrix of a connected graph's incidence matrix is nonsingular if and only if the corresponding edges form a spanning tree. - A circuit matrix represents cycles in a graph, with 1s indicating which edges are in each cycle. - The circuit and incidence matrices of a graph have orthogonal rows, meaning their row dot products are all 0. - A fundamental circuit matrix contains a basis of fundamental cycles, and has an identity submatrix, indicating its rank equals

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Graph Theory: Matrix representation of graphs

The document discusses different matrix representations of graphs:
1) Incidence matrices represent the relationship between vertices and edges, with each column having two 1s. Circuit matrices represent circuits, with each row as a circuit vector. Cut-set matrices represent edge sets whose removal disconnects the graph.
2) Path matrices represent paths between vertex pairs, with columns of all 0s/1s indicating edges not/in every path. Adjacency matrices directly encode vertex connectivity.
3) Exercises are provided to construct the incidence matrix, circuit matrix, fundamental circuit matrix, and cut-set matrix for a given graph.

Graph theory

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.

Isomorphic graph

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 theory presentation

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

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.

Euler graph

The document discusses graphs and Eulerian circuits and paths. It defines what a graph is composed of and defines an Eulerian circuit and path. It states that a graph must be connected and have all vertices visited once for an Eulerian circuit, or have two odd vertices for an Eulerian path. Fleury's algorithm is described for finding an Euler circuit or path by traversing edges without crossing bridges twice. The algorithm works by choosing the next edge such that bridges are only crossed if necessary. Examples are given to demonstrate the algorithm. Applications mentioned include the Chinese postman problem and communicating networks.

Graph representation

A graph consists of vertices and edges, where vertices represent entities and edges represent relationships between vertices. Graphs can be represented sequentially using matrices like adjacency and incidence matrices, or linked using data structures like adjacency lists. Adjacency matrices allow fast addition/removal of edges but use more memory, while adjacency lists use less memory but are slower to modify. The best representation depends on whether the graph is dense or sparse.

Graphs - Discrete Math

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.

Graph Theory: Matrix representation of graphs

The document discusses different matrix representations of graphs:
1) Incidence matrices represent the relationship between vertices and edges, with each column having two 1s. Circuit matrices represent circuits, with each row as a circuit vector. Cut-set matrices represent edge sets whose removal disconnects the graph.
2) Path matrices represent paths between vertex pairs, with columns of all 0s/1s indicating edges not/in every path. Adjacency matrices directly encode vertex connectivity.
3) Exercises are provided to construct the incidence matrix, circuit matrix, fundamental circuit matrix, and cut-set matrix for a given graph.

Graph theory

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.

Isomorphic graph

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 theory presentation

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

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.

Euler graph

The document discusses graphs and Eulerian circuits and paths. It defines what a graph is composed of and defines an Eulerian circuit and path. It states that a graph must be connected and have all vertices visited once for an Eulerian circuit, or have two odd vertices for an Eulerian path. Fleury's algorithm is described for finding an Euler circuit or path by traversing edges without crossing bridges twice. The algorithm works by choosing the next edge such that bridges are only crossed if necessary. Examples are given to demonstrate the algorithm. Applications mentioned include the Chinese postman problem and communicating networks.

Graph representation

A graph consists of vertices and edges, where vertices represent entities and edges represent relationships between vertices. Graphs can be represented sequentially using matrices like adjacency and incidence matrices, or linked using data structures like adjacency lists. Adjacency matrices allow fast addition/removal of edges but use more memory, while adjacency lists use less memory but are slower to modify. The best representation depends on whether the graph is dense or sparse.

Graphs - Discrete Math

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.

Euler paths and circuits

complete guide and training to what is Euler path and circuits with animations and examples
try it to believe it.
made by Shahbaadshah.

Matrix Representation Of Graph

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 theory

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.

Graph isomorphism

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.

Graph theory introduction - Samy

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.

Graph Basic In Data structure

The document provides an overview of graph concepts including definitions of a graph as a pair of vertices and edges, different types of graphs such as directed/undirected graphs and cyclic/acyclic graphs. It also discusses graph coloring, adjacency matrices/lists for representing graphs, and algorithms for traversing graphs including depth-first search and breadth-first search. Examples are given for each concept to illustrate key differences and properties.

Graph theory and life

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.

KARNAUGH MAP(K-MAP)

The document discusses Karnaugh maps (K-maps), which are a tool for representing and simplifying Boolean functions with up to six variables. K-maps arrange the variables in a grid according to their binary values. Adjacent cells that differ in only one variable can be combined to simplify the function by eliminating that variable. The document provides examples of using K-maps to minimize Boolean functions in sum of products and product of sums form. It also discusses techniques like combining cells into the largest groups possible and handling don't-care conditions to further simplify expressions.

Introduction to Graph Theory

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.

Graph Theory: Cut-Set and Cut-Vertices

The document discusses cut-sets and cut-vertices in graphs. It defines a cut-set as a set of edges whose removal disconnects a connected graph. Cut-sets always separate a graph into two disconnected pieces and reduce the graph's rank by one. Theorems are presented regarding the relationship between cut-sets and spanning trees, including that every cut-set must contain at least one branch from every spanning tree. Fundamental cut-sets are also introduced with respect to spanning trees.

Matrix representation of graph

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 .

introduction to 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.

Graph theory

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.

Ppt of graph theory

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.

Graph Theory

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.

Multi ways trees

1. A multi-way search tree allows nodes to have up to m children, where keys in each node are ordered and divide the search space.
2. B-trees are a generalization of binary search trees where all leaves are at the same depth and internal nodes have at least m/2 children.
3. Searching and inserting keys in a B-tree starts at the root and proceeds by comparing keys to guide traversal to the appropriate child node. Insertion may require splitting full nodes to balance the tree.

Graph Theory: Planarity & Dual Graph

Lecture slide for course CSE 4803: Graph Theory
Topics covered:
- Kuratowski's two graphs
- Detection of planarity
- Dual graph
- Dual properties
- Thickness & Crssings
- Completely regular planar graph

Karnaugh map

- Karnaugh maps are used to simplify Boolean algebra expressions by grouping adjacent 1s in a two-dimensional grid.
- Groups must contain powers of 2 cells and cannot include any 0s. They can overlap and wrap around the map.
- The simplified expression is obtained by determining which variables stay the same within each group.

CS6702 Unit III coloring ppt

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.

Graph theory

Basic Graph Theory for Under Graduate level.Easy explanation of all beginner level topic regarding graph theory. I think it will help

Elements of Graph Theory for IS.pptx

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.

Graphs

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.

Euler paths and circuits

complete guide and training to what is Euler path and circuits with animations and examples
try it to believe it.
made by Shahbaadshah.

Matrix Representation Of Graph

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 theory

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.

Graph isomorphism

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.

Graph theory introduction - Samy

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.

Graph Basic In Data structure

The document provides an overview of graph concepts including definitions of a graph as a pair of vertices and edges, different types of graphs such as directed/undirected graphs and cyclic/acyclic graphs. It also discusses graph coloring, adjacency matrices/lists for representing graphs, and algorithms for traversing graphs including depth-first search and breadth-first search. Examples are given for each concept to illustrate key differences and properties.

Graph theory and life

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.

KARNAUGH MAP(K-MAP)

The document discusses Karnaugh maps (K-maps), which are a tool for representing and simplifying Boolean functions with up to six variables. K-maps arrange the variables in a grid according to their binary values. Adjacent cells that differ in only one variable can be combined to simplify the function by eliminating that variable. The document provides examples of using K-maps to minimize Boolean functions in sum of products and product of sums form. It also discusses techniques like combining cells into the largest groups possible and handling don't-care conditions to further simplify expressions.

Introduction to Graph Theory

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.

Graph Theory: Cut-Set and Cut-Vertices

The document discusses cut-sets and cut-vertices in graphs. It defines a cut-set as a set of edges whose removal disconnects a connected graph. Cut-sets always separate a graph into two disconnected pieces and reduce the graph's rank by one. Theorems are presented regarding the relationship between cut-sets and spanning trees, including that every cut-set must contain at least one branch from every spanning tree. Fundamental cut-sets are also introduced with respect to spanning trees.

Matrix representation of graph

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 .

introduction to 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.

Graph theory

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.

Ppt of graph theory

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.

Graph Theory

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.

Multi ways trees

1. A multi-way search tree allows nodes to have up to m children, where keys in each node are ordered and divide the search space.
2. B-trees are a generalization of binary search trees where all leaves are at the same depth and internal nodes have at least m/2 children.
3. Searching and inserting keys in a B-tree starts at the root and proceeds by comparing keys to guide traversal to the appropriate child node. Insertion may require splitting full nodes to balance the tree.

Graph Theory: Planarity & Dual Graph

Lecture slide for course CSE 4803: Graph Theory
Topics covered:
- Kuratowski's two graphs
- Detection of planarity
- Dual graph
- Dual properties
- Thickness & Crssings
- Completely regular planar graph

Karnaugh map

- Karnaugh maps are used to simplify Boolean algebra expressions by grouping adjacent 1s in a two-dimensional grid.
- Groups must contain powers of 2 cells and cannot include any 0s. They can overlap and wrap around the map.
- The simplified expression is obtained by determining which variables stay the same within each group.

CS6702 Unit III coloring ppt

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.

Graph theory

Basic Graph Theory for Under Graduate level.Easy explanation of all beginner level topic regarding graph theory. I think it will help

Euler paths and circuits

Euler paths and circuits

Matrix Representation Of Graph

Matrix Representation Of Graph

Graph theory

Graph theory

Graph isomorphism

Graph isomorphism

Graph theory introduction - Samy

Graph theory introduction - Samy

Graph Basic In Data structure

Graph Basic In Data structure

Graph theory and life

Graph theory and life

KARNAUGH MAP(K-MAP)

KARNAUGH MAP(K-MAP)

Introduction to Graph Theory

Introduction to Graph Theory

Graph Theory: Cut-Set and Cut-Vertices

Graph Theory: Cut-Set and Cut-Vertices

Matrix representation of graph

Matrix representation of graph

introduction to graph theory

introduction to graph theory

Graph theory

Graph theory

Ppt of graph theory

Ppt of graph theory

Graph Theory

Graph Theory

Multi ways trees

Multi ways trees

Graph Theory: Planarity & Dual Graph

Graph Theory: Planarity & Dual Graph

Karnaugh map

Karnaugh map

CS6702 Unit III coloring ppt

CS6702 Unit III coloring ppt

Graph theory

Graph theory

Elements of Graph Theory for IS.pptx

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.

Graphs

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.

ch10.5.pptx

This document section summarizes information about Euler paths and circuits as well as Hamilton paths and circuits. It discusses:
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- Necessary and sufficient conditions for a graph to have an Euler path/circuit
- Algorithms for constructing Euler circuits
- Applications of Euler paths/circuits
- Defining Hamilton paths/circuits and discussing properties like Dirac's and Ore's theorems for Hamilton circuits
- Differences between Euler and Hamilton circuits

Graph Theory,Graph Terminologies,Planar Graph & Graph Colouring

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.

Linear Algebra Gauss Jordan elimination.pptx

Information related to Gauss-Jordan elimination.
Visit mathsassignmenthelp.com or email info@mathsassignmenthelp.com.
You can also call on +1 678 648 4277 for any assistance with linear algebra assignment.

Introduction to-graph-theory-1204617648178088-2

This document provides definitions and theorems related to graph theory. It begins by defining simple graphs, degrees of vertices, and the handshaking lemma. It then discusses paths and cycles in graphs, connectedness, Euler and Hamiltonian paths/circuits. Specific graph types are introduced like trees, planar graphs, and regular graphs. Euler's formula is presented for planar graphs. Definitions of isomorphism and subgraphs are also provided. Theorems regarding trees state that a tree with more than one vertex has at least one vertex of degree one, and that a tree with n vertices has exactly n-1 edges.

graph theory

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.

09_DS_MCA_Graphs.pdf

Graphs can be represented using adjacency matrices or adjacency lists. Common graph operations include traversal algorithms like depth-first search (DFS) and breadth-first search (BFS). DFS traverses a graph in a depth-wise manner similar to pre-order tree traversal, while BFS traverses in a level-wise or breadth-first manner similar to level-order tree traversal. The document also discusses graph definitions, terminologies, representations, elementary graph operations, and traversal methods like DFS and BFS.

Graphs.pdf

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.

Network Topology

This document discusses various network topology concepts including nodes, branches, loops, trees, and different matrix representations of networks. It defines key terms like nodes, branches, loops, meshes, and oriented graphs. It also describes tree concepts such as twigs, links, and co-trees. Finally, it discusses different matrix representations of networks including the incidence matrix, loop matrix, tie-set matrix, cut-set matrix, and formulations of network equilibrium equations in node, mesh, and cut-set forms.

Graphs (Models & Terminology)

This document defines and explains basic graph terminology and representations. It defines simple and directed graphs, and discusses graph models using examples of railway and sports tournament networks. Key graph concepts covered include adjacency, edges, degrees, isolated and pendant vertices, and handshaking theorem. It also explains how to represent graphs using adjacency matrices and incidence matrices.

Network Topology.PDF

This document discusses network topology and its analysis using loop and node variables. It defines key concepts like graphs, trees, co-trees, incidence and tie-set matrices. It shows how to write the equilibrium equations in terms of loop currents using the tie-set matrix. The document also discusses node-pair voltages as network variables and their use in writing nodal analysis equations. Overall, it provides the fundamental concepts and mathematical framework for analyzing electric networks using both loop and node methods.

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This document discusses graphs and graph algorithms. It defines graphs as collections of vertices and edges. It describes different types of graphs like directed, undirected, weighted graphs. It explains graph traversal algorithms like breadth-first search and depth-first search. It also discusses minimum spanning trees and algorithms to find them, like Prim's algorithm and Kruskal's algorithm.

PPT on Graph Theory ( Tree, Cotree, nodes, branches, incidence , tie set and ...

Linear graph, vertex, degree of vertex, subgraph, planar, nonplanar graph, TREES, CO-TREES & LOOPS, complete incidence matrix, reduced incidence matrix, cut set matrix tie set matrix

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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.

Chapter 1

This document provides an overview of key concepts in graph theory, including:
- A graph consists of a set of vertices and edges connecting pairs of vertices.
- Paths and cycles are walks through a graph without repeating edges or vertices. A tree is an acyclic connected graph.
- The degree of a vertex is the number of edges connected to it. Regular graphs have all vertices of the same degree.
- Graphs can be represented using adjacency matrices and incidence matrices to show connections between vertices and edges.
- Directed graphs have edges oriented from a starting to ending vertex. Connectedness in directed graphs depends on the underlying graph or directionality of paths.

Ch08.ppt

The document discusses graphs and their representations. It begins with definitions of simple graphs, directed graphs, and graph terminology such as vertices, edges, degrees, and adjacency. It then covers special graphs like complete graphs, cycles, wheels, and bipartite graphs. The document also discusses operations on graphs like unions and subgraphs. Finally, it introduces ways to represent graphs using adjacency matrices and incidence matrices.

Unit 2: All

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.

Class01_Computer_Contest_Level_3_Notes_Sep_07 - Copy.pdf

The document discusses graph theory concepts including:
- Euler's analysis of the Seven Bridges of Königsberg problem which proved that a walk crossing each bridge once is impossible.
- Definitions of graphs, vertices, edges, degree of a vertex, adjacency matrix, adjacency lists, trees, forests, connectivity, graph traversal algorithms like breadth-first search and depth-first search.
- Explanations and examples are provided for graph representations and common graph terminology.

Bba i-bm-u-2- matrix -

The document discusses various types of matrices:
- Row and column matrices are matrices with only one row or column respectively.
- A square matrix has the same number of rows and columns.
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- 1. SUB MATRICES – CIRCUIT MATRIX (GROUP – 2) MAT2002 – GROUP ACTIVITY
- 2. GROUP MEMBERS ADITI AGRAWAL 20BHI10050 MANAN LADDHA 20BCY10115 RUSHIL BHATNAGAR 20BAI10299 K K ISHVINTHA SREE 20BHI10029 AAKANKSHA PRIYA 20BCY10030 D D
- 3. Subgraph: A graph G’ is said to be a subgraph of a graph G, if all the edges of G’ are in G, and each edge of G’ has the same end vertices in G’ as in G. Let there be a graph G, and let G’ be its subgraph. Let A(G’) and A(G) be the incidence matrices of G’ and G, respectively. Then, A(G’) is a submatrix of A(G). There is a one-to-one correspondence be tween each n by k submatrix of A(G) and a subgraph of G with k edges, k being any positive intege r less than e and n being the number of vertices in G.
- 4. Submatricesof A(G)exhibitspecialproperties Theorem:Let A(G)be an incidencematrixof a connectedgraphG withn vertices. An (n-1)by (n-1)submatrixof A(G)is nonsi ngularif and only if then-1 edgescorrespondingto then-1 columnsof thismatrixconstitutea spanningtreein G. Proof:Everysquare submatrixof ordern-1 in A(G)is thereducedincidencematrixof thesame subgraphin G withn-1 edges, andviceversa. We knowthata square submatrixof A(G)is nonsingularif and onlyif thecorresponding subgraphis a tree. T he treein thiscaseis a spanningtree, because it containsn-1 edgesof then-vertexgraph. Hence, an (n-1)by (n-1)submatrixof an incidence matrixis nonsingularif and only if then-1 edgescorrespondingto then-1 c olumnsof thismatrixconstitutea spanningtreein G.
- 5. Walk & Circuit Walk: A walk is a sequence of vertices a nd edges of a graph i.e. if we traverse a graph then we get a walk. Repetition of vertex and edges is allowed in a walk. . Circuit: A walk that starts and ends at a same vertex and contains no repeated e dges. Note: A circuit is a closed walk, bu t a closed walk need not be a circuit. Closed walk: If a walk starts and ends at the same vertex, then it is said to b e a closed walk. For example, in the figure: Walk: A – B – C – E – A – D , E – D – E – F, etc. Circuit: A – D – E – A , A – B – C – F – E – D – A, etc.
- 6. Circuit Matrix Let the graph G have m edges(e) and let q be the number of different cycles(C) in G. Then the circ uit matrix B = [bij] of G is a (0, 1) − matrix of order q × m, with Forexample:The graph G1 has four different cycles Z1 = {e1, e2}, Z2 = {e3, e5, e7}, Z3 = {e4, e6, e7} and Z4 = {e3, e4, e6, e5}. Its circuitmatrix = B(G1)
- 7. Properties of Circuit Matrix 1 2 3 4 5 A column of all zeros c orresponds to an edge which does not belong to any cycle. Each row of B(G) is a circuit vector. The number of ones in a row is equal to t he number of edges in the corresponding cycle. A circuit matrix has t he property of repre senting a self-loop b y having a single 1 i n the corresponding row. Permutation of any t wo rows or columns in a cycle matrix cor responds to relabeli ng the cycles and th e edges.
- 8. Theorem 7.4 5. If the graph G is separable (or disconnected) and consists of two blocks (or components) H1 and H2, then the cycle matrix B(G) can be written in a block-diagonal form as: ,where B(H1) and B(H2) are the cycle matrices of H1 and H2. STATEMENT • Let B and A be, respectively, the circuit matrix and the incidence matrix (of a self-loop-free graph) whose columns are arranged usi ng the same order of edges. Then every row of B is orthogonal to every row A; that is, A.BT = B.AT = 0 • where the superscripted T denotes a transposed matrix.
- 9. PROOF Consider a vertex v and a circuit ᴦ in the graph G. either v is in ᴦ or it is not. If v is in ᴦ, there ’s no edge in the circuit ᴦ that is incident on v. on the other hand if v is in v, the number of e dges of ᴦ incident on v is exactly two. With this remark in mind, consider the i-th row in and the j-th row in B. Since the edges are arranged in the same order, the non zero entries are in corresponding positions occur only if the particular edge is incident on the i-th vertex and is also in the j-th circuit. If the i-th vertex is not in the circuit, there is no such nonzero entry, and the dot product of the two rows is zero. If the i-th vertex is in the j-th circuit, there will be exactly two 1’s in t he sum of the products of the individual entries. Since 1+1=0, the product of the two arbitrar y rows – one from A and B and other from B – is zero. Hence the theorem is proved. An Exa mple: . 0 0 0 1 0 1 0 1 0 0 0 0 1 1 1 1 0 0 0 0 0 0 0 1 1 1 1 0 1 0 0 0 0 0 1 1 0 0 1 0 1 1 0 0 0 0 0 0 1 0 0 0 1 0 0 0 0 1 0 1 0 0 1 1 0 1 0 1 0 0 1 1 0 1 1 0 0 0 0 0 = [O] = 0
- 10. Fundamental Circuit Matrix and Rank of B • A Submatrix, of a circuit matrix in which all rows correspond to a set of funda mental circuits is called a fundamental circuit matrix Bf. As in matrices A and B, permutations of rows or columns do not affect Bf. If n is the number of ver tices and e is the number of edges in a connected graph, then Bf is (e-n+1) b y e matrix, because the number of fundamental circuits is e-n+1, each fundam ental circuit being produced by one chord. • Let us arrange the columns in Bf such that all the e-n+1 chords correspond t o the first e-n+1 columns. Further let us rearrange the rows such that the firs t row corresponds to the fundamental circuit made by the chord in the first co lumn, the second row to the fundamental circuit made by the second and so o n. This indeed is how the fundamental circuit matrix is arranged in the below fi gure (b)
- 11. e1 e3 e6 e1 e4 e5 e7 1 0 0 1 1 0 1 0 1 0 0 1 0 1 0 0 1 0 0 1 1 e3 e4 e5 e6 Graph and its fundamental circuit matrix (with respect to the spanning tree shown in blue colour) (a) (b)
- 12. A matrix Bf thus arranged can be written as Bf=[Iµ | Bf] - - - - (1) Where, Iµ is an identity matrix of order µ = e-n+1 and Bf is the remaining µ by (n-1) submatrix, corresponding to the branches of the spanning tree. From equation (1) it is clear that the Rank of the Bf = µ = e-n+1 Since Bf is a submatrix of the circuit matrix B, the Rank of B>= e-n+1.
- 13. THANK YOU !