#### Answer

Using the Nearest Neighbor Method, the Hamilton circuit is A,C,D,B,A. The total weight of the circuit is 18.

#### 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 A. We can travel along edge AB, AC, or AD. The weight of these paths are 4, 2, and 4, respectively. We should travel along the edge AC because it has the smallest weight.
From vertex C, we can travel along edge CB or CD. The weights of these paths are 6 and 5 respectively. We should travel along edge CD because it has the smallest weight.
From vertex D, the path must travel to vertex B, and then finally back to vertex A to complete the Hamilton circuit.
Using the Nearest Neighbor Method, the Hamilton circuit is A,C,D,B,A. The edges in this path are AC, CD, DB, and BA. The weights of these edges are 2, 5, 7, and 4.
We can find the total weight of this Hamilton circuit.
total weight = 2 + 5 + 7 + 4
total weight = 18
Using the Nearest Neighbor Method, the Hamilton circuit is A,C,D,B,A. The total weight of the circuit is 18.