Let's address each part of the question step by step.

Part (a): Degree of Each Vertex

To find the degree of each vertex, we count the edges connected to each one.

• A: Connected to B, D, E (3 edges) → Degree = 3
• B: Connected to A, C, D, E (4 edges) → Degree = 4
• C: Connected to B, D, E (3 edges) → Degree = 3
• D: Connected to A, B, C, E (4 edges) → Degree = 4
• E: Connected to A, B, C, D (4 edges) → Degree = 4

Thus, the degrees of each vertex are:

• A: 3
• B: 4
• C: 3
• D: 4
• E: 4

Part (b): Checking for an Euler Circuit or Path

To determine if there is an Euler circuit or path, we apply Euler's theorems:

1. Euler Circuit: Exists if all vertices have even degrees.
2. Euler Path (but not a circuit): Exists if exactly two vertices have odd degrees.

In this graph:

• Vertices A and C have odd degrees (3), while vertices B, D, and E have even degrees (4).
• Since there are exactly two vertices with odd degrees (A and C), an Euler path exists, but not an Euler circuit.

Conclusion:

• There is an Euler path but not an Euler circuit.
• The Euler path would start at one of the odd-degree vertices (A or C) and end at the other.

Would you like more details on finding Euler paths or circuits in different types of graphs?

Further Questions

1. What is the difference between an Euler path and a Hamiltonian path?
2. How do you determine if a graph has a Hamiltonian circuit?
3. Can you have both an Euler circuit and Hamiltonian circuit in the same graph?
4. What happens to the Euler path if we add an edge between two vertices?
5. How does the degree of a vertex affect the structure of a graph?

Tip: When analyzing graphs, start by counting the degrees of vertices to quickly assess properties like Euler paths or circuits.