I found this link that shows you how to make a matrix with random numbers: How to create a matrix with random values in R?

```
#random symmetric matrix with diagnoal elements as 0
A <- matrix(runif(16), 4, 4)
A1 = A %*% t(A)
out_1 = A1 - diag(diag(A1))
[,1] [,2] [,3] [,4]
[1,] 0.00 20371.54 19136.91 18380.49
[2,] 20371.54 0.00 19392.13 19476.78
[3,] 19136.91 19392.13 0.00 16651.73
[4,] 18380.49 19476.78 16651.73 0.00
```

- Imagine each index corresponds to a city in this matrix (out_1) - there are 4 cities.
- Any (row , column) combination corresponds to the distance between two cities (e.g. [2][3] corresponds to the distance between city 2 and city 3).
- Logically, the distance between the same two cities is the same, e.g.. [2] [3] = [3] [2]
- Logically, the distance between a city and itself is 0, e.g. [2] [2] = 0

Using this matrix (out_1), I want to calculate all "paths" from this matrix that have the following structure:

- The path must start and end from the same city, and with the exception of the first city, no other city can be visited twice.

For example, the end goal would look something like this:

```
- Path 1 : [1] [2] , [2] [3], [3] [4], [4] [1] = 20371.54 + 19392.13 + 16651.73 + 18380.49 = 74795.89
- Path 2 : [2] [4] , [4] [3], [3] [1], [1] [2] = 19476.78 + 16651.73 + 19136.91 + 20371.54 = 75636.96
- ....
- Path n: [3] [4] , [4] [1], [1] [2], [2] [3] = 16651.73 + 18380.49 + 20371.54 + 19392.13 = 74795.89
```

I tried to find some library in R to do this (note: I think what I am trying to do is called a "Hamiltonian Cycle of a Graph") : hamiltonian function - RDocumentation

But this did not work:

```
c = c(out_1)
hamiltonian(c, cycle = TRUE)
Error in graph_vect2list(edges) :
Argument 'edges' is not a correct edge list.
```

How can I get a list of all such "paths" that shows the order of the cities and the total distance for each path?

Thanks!