This was a relatively simple problem. I was happy to see my 10 year old, Arushi, solve this one. I also got correct answers from Karan Sharma, Mahi Saraf, Anahat Prakash, Pooja Goyal, Pratik Poddar and Suman Saraf. Well done everyone!
There are two possible solutions, though both have the same basis and are just vertically flipped solutions. Here is the answer:
Solution to Puzzle #183: 8 Digits
Here is a quick explanation, copied from Pratik Poddar:
Alok Goyal's Puzzles Page 
C and F independently are connected to all except one circle. So, C and F cannot have two adjacent numbers. So C is 1 and F is 8. So A is 7 and H is 2.
3,4,5,6 needs to be placed in B,D,E,G
B and D cannot be 6
E and G cannot be 3
Since the image is symmetric across vertical line, lets say B is 3. If E is 6, and D and G become 4 and 5 which is not possible. So G is 6. So D is 4 and E is 5.
So, A is 7, B is 3, C is 1, D is 4, E is 5, F is 8 and G is 6.
Hope you all enjoyed the puzzle!