The incidence matrix $A(G)$ of a graph $G$ is a matrix that represents the relationship between vertices and edges. The rank of the incidence matrix is related to the structure of the graph, particularly to its connected components.

Theorem:

If $G$ is a graph with $n$ vertices and $k$ connected components, then the rank of its incidence matrix $A(G)$ is $n - k$.

Explanation:

The rank of the incidence matrix $A(G)$ is $n - k$, where:

• $n$ is the number of vertices,
• $k$ is the number of connected components.

This is because in each connected component of the graph, there is one "degree of freedom" (since every connected component has one fewer independent edge than vertices). Therefore, for each connected component, the incidence matrix loses one rank, leading to a total reduction of $k$ in the rank.

Example:

Consider a graph $G$ with 5 vertices and 2 connected components.

1. Vertices: $V = \{1, 2, 3, 4, 5\}$.
2. Edges:
   • Component 1: $e_1 = (1, 2), e_2 = (2, 3)$,
   • Component 2: $e_3 = (4, 5)$.

The incidence matrix $A(G)$ looks like this:

\begin{bmatrix} 1 & 0 & 0 \\ -1 & 1 & 0 \\ 0 & -1 & 0 \\ 0 & 0 & 1 \\ 0 & 0 & -1 \end{bmatrix}$$ #### Step 1: Check the number of vertices and components - $$n = 5$$ (5 vertices: $$\{1, 2, 3, 4, 5\}$$). - $$k = 2$$ (2 connected components). #### Step 2: Find the rank of the incidence matrix For a graph with $$k$$ connected components, the rank of the incidence matrix $$A(G)$$ is $$n - k = 5 - 2 = 3$$. ### Conclusion: The rank of the incidence matrix $$A(G)$$ in this example is $$3$$, which corresponds to $$n - k = 5 - 2$$. This illustrates that for a graph with $$n$$ vertices and $$k$$ connected components, the rank of its incidence matrix is $$n - k$$. --- Would you like further details on this topic or additional examples? ### 5 Related Questions: 1. How is the rank of the incidence matrix computed for a cycle graph? 2. What is the relationship between the nullity of the incidence matrix and the number of connected components? 3. How does the rank of the adjacency matrix compare to the incidence matrix? 4. Can you explain the rank of the incidence matrix for a complete graph? 5. How does the incidence matrix change if you add more edges between components? ### Tip: When calculating the rank of an incidence matrix, it's helpful to remember that every connected component contributes to reducing the rank by exactly one.