#### Answer

Using the Nearest Neighbor Method, the Hamilton circuit is Home,Bank,Post Office,Market,Home. The total weight of the circuit is 15. Since the weights represent miles, the total distance for this Hamilton circuit is 15 miles. This is the same result that we found using the Brute Force Method.

#### Work Step by Step

With the Nearest Neighbor Method, we should choose the edge which has the smallest weight for each step along the path.
Let's start at vertex Home. We can travel along the edge Home-Bank, Home-Post Office, or Home-Market. The weights of these edges are 3, 5.5, and 3.5, respectively. We should travel along the edge Home-Bank because it has the smallest weight.
From vertex Bank, we can travel along edge Bank-Post Office or Bank-Market. The weights of these paths are 4 and 5, respectively. We should travel along edge Bank-Post Office because it has the smallest weight.
From vertex Post Office, the path must travel to vertex Market, and then finally back to vertex Home to complete the Hamilton circuit.
Using the Nearest Neighbor Method, the Hamilton circuit is Home,Bank,Post Office,Market,Home. The weights of the edges in this path are 3, 4, 4.5, and 3.5.
We can find the total weight of this Hamilton circuit.
total weight = 3 + 4 + 4.5 + 3.5
total weight = 15
Using the Nearest Neighbor Method, the Hamilton circuit is Home,Bank,Post Office,Market,Home. The total weight of the circuit is 15. Since the weights represent miles, the total distance for this Hamilton circuit is 15 miles. This is the same result that we found using the Brute Force Method.