# How to predict all non-isomorphic connected simple graphs are there with $n$ vertices

$$(1)\:$$How many non-isomorphic connected simple graphs are there with $$n$$ vertices when n is,
$$\qquad(a)\:4\qquad(b)\:5$$
$$(2)\:$$Draw all non-isomorphic, cycle free, connected graphs having six vertices.

For $$(1)$$ when $$n=4$$, it's only $$6$$ case I got. But when it is $$5$$, I was unable to find out $$21$$ case. Actually I found $$15$$ case and here is my question arrive,

Is there any prediction without drawing all of those case$$?$$ If not then how could someone ensure his/her answer in Exam $$(\text{For big enough n})?$$

For $$(2)$$ I have the same situation.
I was thinking there should be other way to predict the answer of this kind question. Any help will be appreciated.
• Seems like induction on the maximum degree of a vertex ought to go a long way in the cycle-free case. If $n=4$, I can only find two connected cycle-free graphs. – Robert Shore Dec 30 '19 at 23:25
• @MarkoRiedel It seems to understand your answer$(\text{Linked})$ I need many stuff to study. But at that time it will be best if I have a generating function/recurrence relation to get my desire answer. Can you provide any$?$ And thanks for your response I will definitely read your answer$(\text{Linked})$ after exam. – NajmunNahar Dec 31 '19 at 7:10
• @MarkoRiedel Any OEIS sequence which can tell for $n=1,\cdots,10?$ – Dr.Antidode Dec 31 '19 at 9:11
I tried for $$(2)$$ and got $$6$$ case$$(\text{Sorry for my poor drawing})$$, 