Let's solve the problem step-by-step.

Part (a): Determine the Degree of Each Vertex

In graph theory, the degree of a vertex is the number of edges connected to it. Let’s calculate the degree for each vertex in the provided graph.

From visual inspection:

• Vertex A has edges connecting to B, C, D, and E, so its degree is 4.
• Vertex B has edges connecting to A, C, and D, so its degree is 3.
• Vertex C has edges connecting to A, B, and E, so its degree is 3.
• Vertex D has edges connecting to A, B, and E, so its degree is 3.
• Vertex E has edges connecting to A, C, and D, so its degree is 3.

So, the degrees are:

• A: 4
• B: 3
• C: 3
• D: 3
• E: 3

Part (b): Euler Circuit or Path

An Euler circuit exists if all vertices have an even degree, and an Euler path exists if exactly two vertices have an odd degree.

In this graph:

• Vertex A has an even degree (4).
• Vertices B, C, D, and E all have an odd degree (3).

Since there are four vertices with odd degrees, this graph does not have an Euler circuit or an Euler path.

Conclusion:

1. Degrees of vertices: A: 4, B: 3, C: 3, D: 3, E: 3.
2. No Euler circuit or path exists because there are more than two vertices with an odd degree.

Would you like further clarification on any of these points?

Related Questions

1. What is the difference between an Euler circuit and an Euler path?
2. Can this graph be modified to have an Euler path by adding or removing edges?
3. How would you find a Hamiltonian path in this graph, if one exists?
4. What other types of circuits or paths can exist in graphs with mixed-degree vertices?
5. How does the concept of degrees relate to bipartite graphs?

Tip:

In graph theory, knowing the degree of each vertex quickly helps in identifying possible Eulerian paths and circuits.