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Graph theory

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This document summarizes different graph theory concepts and methods for solving systems of linear and nonlinear equations using graph theory. It defines what graph theory is, provides examples of different graph types, and discusses the Laplacian matrix. It also outlines Gaussian elimination and other methods for solving linear systems, as well as Newton's method and the secant method for nonlinear systems. An example using Gaussian elimination to solve a system of 3 linear equations is also included.

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Solution of System of Linear and Non Linear Equations Using Graph Theory Team Members: Amar Chand : 13112009 Deeksha Adlakha : 13112028
What is Graph Theory? Graph theory is the study of graphs, which are mathematical structures used to model pairwise relations between objects. V = {a, b, c, d, e} E = {ab, ac, bd, cd, de}
Applications of Graph Theory • Computer Science • Electrical Engineering • Chemical Science • General • Other
Types of Graphs Null Graph Trivial Graph • a Non-Directed GraphDirected Graph Simple Graph
Connected Graph Disconnected Graph Regular Graph Complete Graph
Laplacian matrix A simple graph G with n vertices, its Laplacian matrix L is defined as : L = D - A where D is the degree matrix A is the adjacency matrix of the graph. The elements of L are given by : where Deg(Vi) is degree of the vertex i. Li,j =
Example :-
Methods for solving a system of equations Linear Equations: 1. Gaussian Elimination Method 2. Gauss Jordan Elimination Method 3. Gauss Seidel Method
Non Linear Equations: 1. Newton’s Method 2. Secant Method
Gaussian Elimination Method Algorithm for solving system of linear equations (Ax=b) by bringing the augmented matrix to an upper triangular form.
Example Solve the following system of linear equations to obtain the values of x,y,z using Gaussian Elimination Method . x – 3y +z = 4 2x – 8y + 8z = -2 -6x + 3y – 15z = 9
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Graph theory
Graph theory
Graph theory
Graph theory
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Graph theory

  • 1. Solution of System of Linear and Non Linear Equations Using Graph Theory Team Members: Amar Chand : 13112009 Deeksha Adlakha : 13112028
  • 2. What is Graph Theory? Graph theory is the study of graphs, which are mathematical structures used to model pairwise relations between objects. V = {a, b, c, d, e} E = {ab, ac, bd, cd, de}
  • 3. Applications of Graph Theory • Computer Science • Electrical Engineering • Chemical Science • General • Other
  • 4. Types of Graphs Null Graph Trivial Graph • a Non-Directed GraphDirected Graph Simple Graph
  • 5. Connected Graph Disconnected Graph Regular Graph Complete Graph
  • 6. Laplacian matrix A simple graph G with n vertices, its Laplacian matrix L is defined as : L = D - A where D is the degree matrix A is the adjacency matrix of the graph. The elements of L are given by : where Deg(Vi) is degree of the vertex i. Li,j =
  • 8. Methods for solving a system of equations Linear Equations: 1. Gaussian Elimination Method 2. Gauss Jordan Elimination Method 3. Gauss Seidel Method
  • 9. Non Linear Equations: 1. Newton’s Method 2. Secant Method
  • 10. Gaussian Elimination Method Algorithm for solving system of linear equations (Ax=b) by bringing the augmented matrix to an upper triangular form.
  • 11. Example Solve the following system of linear equations to obtain the values of x,y,z using Gaussian Elimination Method . x – 3y +z = 4 2x – 8y + 8z = -2 -6x + 3y – 15z = 9