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### Question 5: Graph Representation with Adjacency Matrix and Incidence Matrix

Let's first represent the graphs \(K_{1,4}\) and \(W_4\) with both adjacency matrices and incidence matrices.

#### 1. **Graph \(K_{1,4}\):**

**Adjacency Matrix:**

\(K_{1,4}\) is a star graph where one vertex is connected to 4 others. Suppose vertex 1 is the central vertex connected to vertices 2, 3, 4, and 5.

\[
\begin{pmatrix}
0 & 1 & 1 & 1 & 1 \\
1 & 0 & 0 & 0 & 0 \\
1 & 0 & 0 & 0 & 0 \\
1 & 0 & 0 & 0 & 0 \\
1 & 0 & 0 & 0 & 0 \\
\end{pmatrix}
\]

**Incidence Matrix:**

If the edges are represented as \(e_1, e_2, e_3, e_4\) (with each connecting the central vertex to another), the incidence matrix would look like this:

\[
\begin{pmatrix}
1 & 1 & 1 & 1 \\
1 & 0 & 0 & 0 \\
0 & 1 & 0 & 0 \\
0 & 0 & 1 & 0 \\
0 & 0 & 0 & 1 \\
\end{pmatrix}
\]

#### 2. **Graph \(W_4\):**

\(W_4\) is a wheel graph consisting of a cycle with 4 vertices and an additional central vertex connected to all others.

**Adjacency Matrix:**

Assume the vertices are labeled such that 1 is the central vertex and 2, 3, 4, 5 form a cycle.

\[
\begin{pmatrix}
0 & 1 & 1 & 1 & 1 \\
1 & 0 & 1 & 0 & 1 \\
1 & 1 & 0 & 1 & 0 \\
1 & 0 & 1 & 0 & 1 \\
1 & 1 & 0 & 1 & 0 \\
\end{pmatrix}
\]

**Incidence Matrix:**

If we label the edges of the cycle as \(e_1, e_2, e_3, e_4\) and the spokes (connections to the center) as \(e_5, e_6, e_7, e_8\), the incidence matrix is:

\[
\begin{pmatrix}
1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 \\
0 & 1 & 0 & 0 & 1 & 0 & 0 & 1 \\
1 & 0 & 1 & 0 & 0 & 1 & 0 & 0 \\
0 & 1 & 0 & 1 & 0 & 0 & 1 & 0 \\
1 & 0 & 1 & 0 & 0 & 0 & 0 & 1 \\
\end{pmatrix}
\]

### Question 6: **Euler Circuit**

**Definition:**
An **Euler circuit** is a circuit that uses every edge of a graph exactly once and returns to the starting vertex. An **Euler path** uses every edge exactly once but does not necessarily return to the starting vertex.

For a graph to have an **Euler circuit**, all vertices must have an even degree. For an **Euler path**, exactly two vertices should have an odd degree.

**Determination for the Given Graph:**

1. **Degrees of Vertices:**
   - \( \deg(a) = 3 \)
   - \( \deg(b) = 3 \)
   - \( \deg(c) = 3 \)
   - \( \deg(d) = 3 \)
   - \( \deg(e) = 4 \)

Since all vertices except \(e\) have odd degrees, the graph cannot have an **Euler circuit**.

2. **Euler Path:**
   Since more than two vertices have odd degrees (in fact, four vertices), the graph does not have an **Euler path** either.

### Summary:
- **Graph \(K_{1,4}\)** and **\(W_4\)** are represented by their respective adjacency and incidence matrices.
- The graph in **Question 6** does not have an Euler circuit or path, as it does not satisfy the degree conditions required for either.

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Would you like a deeper dive into any of these solutions or additional details? Here are some related questions:

1. What is the difference between a Hamiltonian cycle and an Euler circuit?
2. Can a bipartite graph have an Euler circuit?
3. How do you construct the incidence matrix of a directed graph?
4. What are the conditions for a connected graph to have an Eulerian trail?
5. How do you convert a graph's adjacency matrix to its incidence matrix?

**Tip:** When analyzing a graph for an Euler circuit or path, always start by checking the degrees of all vertices.

5. Represent each of these graphs with an adjacency matrix and incidence matrix: K_{1,4}, W_4.
6. Define Euler circuit. Determine whether the following graph has an Euler circuit or Euler path and construct such a circuit or path, if it exists.

Learn how to represent the graphs K_{1,4} and W_4 using adjacency and incidence matrices. Analyze the given graph to determine whether it has an Euler circuit or Euler path, based on vertex degrees. This problem covers key concepts in graph theory, including Euler's theorem.

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