#### Answer

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

#### Work Step by Step

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