There is a question in the book "Discrete Mathematics and Applications" by Susanna Epp in the exercise set $10.4$ :
$20.$ Draw four nonisomorphic graphs with six vertices, two of degree $4$ and four of degree $3$.
[Note : By Graphs we mean undirected graphs where multiple edges and loops are allowed.]
I used handshaking theorem to find number of edges.
$\therefore 2(\text{Number of edges})=(4+4+3+3+3+3) \implies (\text{Number of edges})=10.$
I drew four non-isomorphic graphs as asked as follows :
After drawing them I found a fifth one as well:
Perhaps there are two more that I found out later.
Which led me to think whether there are counting techniques to count total number of non-isomorphic graphs for this problem?
EDIT : Like this post of mine, aren't there any smooth ways? In that post I took cases having one circuit, two circuit,...etc which made it simple to count all the non-isomorphic graphs.