I get 2 and 4 respectively. How did you get 6 and 10? Can you give an example of two vertices that are six apart in G1 or 10 apart in G2? The diameter is the longest shortest path. In G1, the distance from vertices 1 to 2 is 1. You could go 1-5-4-9-6-8-10-7-2 (a path of length 8), but that is not the distance between vertices 1 and 2. The distance is the length of the shortest path, which is 1. The diameter of the graph is the largest distance, given all distances between every two vertices.
Let's look at distances from vertex 1:
1-2 = 1
1-2-3 = 2
1-5-4 = 2
1-5 = 1
1-6 = 1
1-2-7 = 2
1-6-8 = 2
1-6-9 = 2
1-5-10 = 2
By symmetry, this is the same as the distances for paths starting at 2, 3, 4, and 5. It is also the same for paths starting from the inner vertices to the outer vertices. We still need to check from the inner vertices to each other:
6-9-7 = 2
6-8 = 1
6-9 = 1
6-8-10 = 2
By symmetry, this is the same as the distance from any inner vertex to any other inner vertex. So, the maximum distance between any two vertices will be 2.
For G2, it is a little more difficult. From 1 to 11, the maximum distance that I could find was 4. I do not see any two vertices that would have a longer distance between them. There are not as many symmetries in this graph, so to brute force it, you would need to check the distance between every two vertices. Since there are 11 vertices, there are only $\dbinom{11}{2}=55$ different pairs to check distances. I believe you will find the answer to be 4, but I have not gone through them all.
Thanks SlipEternal ,with G1 I wasn't using paths directly and just making my own straight lines from the different vertices as the maximum distance between two vertices and interpreted the meaning of how to do them completely wrong
for example I would go 1-3 =1 2-4 =1 3-5=1
1-4 =1 2-5 =1 3-1= 1 in total adding to 6.
I worked in a clockwise direction until I reached vertice 1 again .
When our lecturer put the one example on the board it kind of looked like he did it this way and I think I was confused .
Similar case for G2 ...Oh dear I will definitely need some practice with these .
Your explanation is very good and I can make sense of it all .They tend to go very very quickly in the lectures and then we move on to another topic ..
Thanks again SlipEternal and very much appreciated .
Cheers