Consider the complete graph with 5 vertices, denoted by K5.

A. Draw the graph.

B. How many edges are in K5? How many edges are in Kn?

C. Find an isomorphic representation (graph) of K5. Give the isomorphism mappings.

D. Does K5 contain Eulerian circuits? (why?) If yes, draw them.

E. Does K5 contain Hamiltonian circuits? If yes, draw them.

F. How many walks of length 2 are there in graph K5? Explain.

A. The complete graph with 5 vertices, K5, can be drawn as follows:

- Each of the 5 vertices is represented as a point.
- Every pair of vertices is connected by an edge.

B. The number of edges in K5 can be calculated using the formula:

Number of edges in K5 = (n * (n-1))/2

where n is the number of vertices. Plugging in n = 5, we get:

Number of edges in K5 = (5 * (5-1))/2 = 10

For Kn, where n is the number of vertices, the number of edges can also be calculated using the same formula. So, the number of edges in Kn would be (n * (n-1))/2.

C. An isomorphic representation of K5 can be obtained by relabeling the vertices of the original graph. In K5, all vertices are connected to each other, so there are no differences between the vertices. Thus, any permutation of the vertex labels will result in an isomorphic graph.

Here is one possible isomorphism mapping for K5:

Original labeling: A, B, C, D, E
Isomorphism labeling: E, A, D, B, C

D. An Eulerian circuit is a path that traverses each edge of a graph exactly once. In order for a graph to have an Eulerian circuit, all of its vertices must have an even degree.

In the case of K5, each vertex is connected to four other vertices, resulting in all vertices having an odd degree. Therefore, K5 does not contain any Eulerian circuits.

E. A Hamiltonian circuit is a path that visits each vertex of a graph exactly once. K5, being a complete graph, has every pair of vertices connected. Therefore, we can find a Hamiltonian circuit in K5.

One possible Hamiltonian circuit in K5 can be described as follows:
Start at any vertex. Move to an adjacent vertex. Continue this process until you have visited all vertices, and then return to the starting vertex.

F. To find the number of walks of length 2 in graph K5, we need to count the number of possible paths that consist of 2 edges.

Since K5 is a complete graph, it means that every pair of vertices is connected by exactly one edge. So, for any given vertex, we can choose any of the other 4 vertices to create a two-edge walk.

Since there are 5 vertices in total, we can choose the starting vertex in 5 ways. For each starting vertex, we can then choose the next vertex in 4 ways. Therefore, the total number of walks of length 2 in K5 is 5 * 4 = 20.