Networks & Matrices (Edexcel A-Level Further Mathematics): Revision Notes

Example: Adjacency Matrix for an Undirected Graph

For a graph with vertices $A, B, C, D$

Edges: $AB, BC, CD, DA$

Example: Weighted Matrix

For a graph with $A, B, C$:

• $AB = 5, BC = 3, AC = \infty$

Example:

Given the matrix:

Steps to Draw the Network:

• Vertices: $A, B, C, D$
• Edges:
• $AB$ (weight = 7), $AC$ (weight = $2$).
• $BD$ (weight = 3), $CD$ (weight = $6$).

Example:

For the graph with:

• Vertices: $A, B, C, D$
• Edges: $AB = 7, AC = 2, BD = 3, CD = 6$

The weighted matrix is:

Example 1: Drawing a Network from a Weighted Matrix

Given:

Steps:

1. Label vertices $A, B, C$
2. Add edges:

• $AB$ with weight $4$.
• $BC$ with weight $5$

Network:

• Vertices: $A, B, C$
• Edges: $AB = 4, BC = 5$

Example 2: Writing a Weighted Matrix from a Network

Given the graph:

• Vertices: $A, B, C, D$
• Edges: $AB = 3, AC = 6, AD = 4, BC = 5$

Steps:

3. Assign rows/columns for vertices.
4. Fill weights:

Common Mistakes

1. Misinterpreting weights: Ensure weights in a matrix correspond to correct edges.
2. Ignoring symmetry: For undirected graphs, matrices must be symmetric ($a_{ij} = a_{ji}$).
3. Confusing zero entries: A zero entry often means no edge, except on the diagonal where it means no self-loop.
4. Mislabeling vertices: Ensure consistency between vertex labels and matrix rows/columns.
5. Forgetting edge directions: For directed graphs, entries may not be symmetric.

Key Formulas and Concepts

1. Adjacency Matrix:

2. Weighted Matrix: