#### Answer

Using the Nearest Neighbor Method, the Hamilton circuit is A,B,E,D,C,A. The total weight of the circuit is 24.

#### 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, AD, or AE. The weights of these paths are 4, 7, 5, and 10, respectively. We should travel along the edge AB because it has the smallest weight.
From vertex B, we can travel along edge BC, BD, or BE. The weights of these paths are 7, 9, and 6, respectively. We should travel along edge BE because it has the smallest weight.
From vertex E, we can travel along edge EC or ED. The weights of these paths are 8 and 4, respectively. We should travel along edge ED because it has the smallest weight.
From vertex E, the path must travel to vertex C, and then finally back to vertex A to complete the Hamilton circuit.
Using the Nearest Neighbor Method, the Hamilton circuit is A,B,E,D,C,A. The edges in this path are AB, BE, ED, DC, and CA. The weights of these edges are 4, 6, 4, 3, and 7.
We can find the total weight of this Hamilton circuit.
total weight = 4 + 6 + 4 + 3 + 7
total weight = 24
Using the Nearest Neighbor Method, the Hamilton circuit is A,B,E,D,C,A. The total weight of the circuit is 24.