**Solution:**

To find the resistance between two diagonally opposite corners of the cube, we can consider the cube to be made up of three sets of parallel resistors. Let's break down the solution into steps:

**Step 1:**
First, let's identify the three sets of parallel resistors in the cube:

- Set 1: Contains the three resistors along one edge of the cube.
- Set 2: Contains the three resistors along the edge perpendicular to Set 1.
- Set 3: Contains the three resistors along the remaining edge of the cube.

**Step 2:**
Next, let's calculate the equivalent resistance for each set of parallel resistors:

- For Set 1, since the three resistors are in parallel, the equivalent resistance is given by the formula: 1/Req = 1/R1 + 1/R2 + 1/R3
- Since all the resistors are identical with resistance 5ohm, we can plug in the values and calculate the equivalent resistance: 1/Req = 1/5 + 1/5 + 1/5 = 3/5
- Taking the reciprocal of both sides, we find that the equivalent resistance for Set 1 is 5/3ohm.

**Step 3:**
Similarly, for Set 2 and Set 3, the equivalent resistance can be calculated as 5/3ohm.

**Step 4:**
Now, let's analyze the connections between the three sets of parallel resistors:

- The resistors in Set 1 and Set 2 are connected in series, so their equivalent resistance is the sum of their individual resistances: Req1+2 = 5/3 + 5/3 = 10/3ohm.
- The resistors in Set 1 and Set 3 are also connected in series, so their equivalent resistance is the sum of their individual resistances: Req1+3 = 5/3 + 5/3 = 10/3ohm.
- Finally, the resistors in Set 2 and Set 3 are connected in parallel, so their equivalent resistance is given by: 1/Req2+3 = 1/(5/3) + 1/(5/3) + 1/(5/3) = 3/5 + 3/5 + 3/5 = 9/5.
- Taking the reciprocal of both sides, we find that the equivalent resistance for the combination of Set 2 and Set 3 is 5/9ohm.

**Step 5:**
Now, the resistors in Set 1+2 and Set 2+3 are connected in series, so their equivalent resistance is the sum of their individual resistances:

- Req1+2+2+3 = Req1+2 + Req2+3 = 10/3 + 5/9 = (30 + 5)/9 = 35/9ohm.

**Step 6:**
Finally, the two diagonally opposite corners of the cube are connected by the resistors in Set 1+2+2+3, so the resistance between these two corners is equal to the equivalent resistance of Set 1+2+2+3, which is 35/9ohm.

Therefore, the correct option is 1) 35/9