1. What is the Rank of Incidence Matrix?

The rank of an incidence matrix in graph theory represents the number of linearly independent rows or columns. For a connected graph with N nodes, the rank of the incidence matrix is always N-1.

2. How Does the Calculator Work?

The calculator uses the simple formula:

Where:

• \( \rho \) — Matrix rank
• \( N \) — Number of nodes in the graph

Explanation: This formula applies to connected graphs where the incidence matrix represents the relationship between nodes and edges.

3. Importance of Matrix Rank Calculation

Details: The rank of incidence matrix is crucial in graph theory, network analysis, and electrical circuit analysis as it determines the number of independent equations needed to describe the system.

4. Using the Calculator

Tips: Enter the number of nodes in the graph. The value must be a positive integer greater than 0.

5. Frequently Asked Questions (FAQ)

Q1: Why is the rank always N-1 for connected graphs?
A: Because in a connected graph, there is always one linear dependency among the rows/columns of the incidence matrix.

Q2: Does this formula work for disconnected graphs?
A: No, for disconnected graphs with k components, the rank would be N - k.

Q3: What is an incidence matrix?
A: An incidence matrix is a matrix that shows the relationship between vertices (nodes) and edges in a graph.

Q4: Where is this concept applied?
A: This concept is widely used in electrical engineering, computer networks, and graph theory applications.

Q5: Can the rank be greater than N-1?
A: No, for any connected graph with N nodes, the maximum rank of the incidence matrix is always N-1.