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.

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 =

7. Example :-

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

12. C++ Code

17. THANKS!