#### Answer

The smallest total weight is 18 when the Hamilton circuit is A,B,D,C,A or A,C,D,B,A.

#### Work Step by Step

The optimal solution is the Hamilton circuit with the smallest total weight. We can use the Brute Force Method to find the optimal solution. The Brute Force Method involves listing all the possible Hamilton circuits and calculating the total weight of each Hamilton circuit.
To find the total weight of a path, we can simply add up the weight of each edge in the path.
The Hamilton circuit is A,B,C,D,A.
total weight = 4 + 6 + 5 + 4 = 19
The Hamilton circuit is A,B,D,C,A.
total weight = 4 + 7 + 5 + 2 = 18
The Hamilton circuit is A,C,B,D,A.
total weight = 2 + 6 + 7 + 4 = 19
The Hamilton circuit is A,C,D,B,A.
total weight = 2 + 5 + 7 + 4 = 18
The Hamilton circuit is A,D,B,C,A.
total weight = 4 + 7 + 6 + 2 = 19
The Hamilton circuit is A,D,C,B,A.
total weight = 4 + 5 + 6 + 4 = 19
We can see that the smallest total weight is 18 when the Hamilton circuit is A,B,D,C,A or A,C,D,B,A.