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$(1)\:$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.
Thanks in advances.

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  • $\begingroup$ The following MSE link on non-isomorphic connected graphs might prove useful. $\endgroup$ – Marko Riedel Dec 30 '19 at 23:17
  • $\begingroup$ 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. $\endgroup$ – Robert Shore Dec 30 '19 at 23:25
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    $\begingroup$ @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. $\endgroup$ – NajmunNahar Dec 31 '19 at 7:10
  • $\begingroup$ @MarkoRiedel Any OEIS sequence which can tell for $n=1,\cdots,10?$ $\endgroup$ – Dr.Antidode Dec 31 '19 at 9:11
  • $\begingroup$ oeis but there does not seem to be any generating function of formula. $\endgroup$ – almagest Dec 31 '19 at 11:33
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I tried for $(2)$ and got $6$ case$(\text{Sorry for my poor drawing})$, enter image description here
I think the main hack will be to find all possible tree.

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