Let's analyze and solve the questions one by one:

1. Total Degree:

   • The total degree of a graph is the sum of the degrees of all vertices. In an undirected graph, the total degree is twice the number of edges.
   • Given that the graph $G$ has 8 edges, the total degree is: $2 \times 8 = 16$
   • Answer: Total degree = 16.

2. Degree of a One Vertex:

   • There are 4 vertices in the graph, and each vertex has the same degree.
   • Since the total degree is 16, each vertex has a degree of: $\frac{16}{4} = 4$
   • Answer: Degree of one vertex = 4.

3. Does an Euler Circuit Exist in $G$?

   • An Euler Circuit exists if the graph is connected and every vertex has an even degree.
   • Here, each vertex has a degree of 4, which is even. If we assume the graph is connected (which is often implied unless stated otherwise), then an Euler Circuit exists.
   • Answer: Yes, an Euler Circuit exists.

4. Does an Euler Path Exist in $G$?

   • An Euler Path exists if the graph is connected and has exactly zero or two vertices with odd degrees.
   • Since every vertex has an even degree, this condition is met as well.
   • Answer: Yes, an Euler Path exists (an Euler Circuit also implies the existence of an Euler Path).

5. Number of Components of $G$:

   • For an Euler Circuit to exist, the graph must be connected, meaning there is only one component.
   • Answer: Number of components of $G$ = 1.

Let me know if you’d like further explanation on any of these answers, or if you have any additional questions.

Here are five questions related to this topic:

1. What conditions are necessary for a graph to have an Euler Path but not an Euler Circuit?
2. How would the answers change if the graph had an odd number of edges?
3. Can a graph with a different number of components have an Euler Circuit?
4. How would the presence of more or fewer vertices affect the degree of each vertex in a similar setup?
5. What is the difference between a Hamiltonian Circuit and an Euler Circuit?

Tip: In graph theory, always check the degree of vertices to determine if Euler paths or circuits are possible, as this simplifies the analysis.