#### Answer

Using the Nearest Neighbor Method, the Hamilton circuit is A,E,D,C,B,A. The total weight of the circuit is 33.

#### 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 11, 6, 8, and 5, respectively. We should travel along the edge AE because it has the smallest weight.
From vertex E, we can travel along edge BE, CE, or DE. The weights of these paths are 10, 9, and 8, respectively. We should travel along edge DE because it has the smallest weight.
From vertex D, we can travel along edge BD or CD. The weights of these paths are 7 and 4, respectively. We should travel along edge CD because it has the smallest weight.
From vertex C, 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,E,D,C,B,A. The edges in this path are AE, DE, CD, BC, and AB. The weights of these edges are 5, 8, 4, 5, and 11.
We can find the total weight of this Hamilton circuit.
total weight = 5 + 8 + 4 + 5 + 11
total weight = 33
Using the Nearest Neighbor Method, the Hamilton circuit is A,E,D,C,B,A. The total weight of the circuit is 33.