Describe all bipartite graphs modulo isomorphism with 12 vertices and each vertex of degree 3
Hi. I've been trying to figure what this question asks. Is this asking to draw and find out all graphs with 12vertices those are bipartite graphs and isomorphic among each other? How is "modulo isomorphism" different from "isomorphism?
I've been thinking about the question in your last comment, and I believe I understand it now.
This isn't a solution, but I don't think I can fit my response in a comment box. We have $12$ vertices of degree $3,$ so we know that the graph must have $18$ edges. Then there can be at most $6$ vertices in a bipartition set, since that will account for all $18$ edges. Therefore, both bipartition sets contain $6$ vertices. Call these sets $U$ and $V$.
The graph can be determined by its adjacency matrix. This is a $12\times12$ symmetric matrix of $0$s and $1$s. If we list all the elements of $U$ before those of $V$ then the adjacency matrix takes the form $$\begin{bmatrix}0&A\\A^T&0\end{bmatrix}$$ where 0 indicates a $6\times6$ zero matrix and $A$ is $6\times6$ matrix with three $0$s and three $1$s in each row and each column.
So, we can approach the problem by trying to determine all such matrices A. If you were to do it this way (and I'm not saying this is the best approach) then yes, you would no doubt generate many isomorphic graphs, and part of the problem would be to separate them into equivalence classes, and describe or draw one representative of each.
I don't know whaat the best way to approach this problem is. If you do use the approach alluded to above, be sure to apply as much symmetry as you can. For example, it's clear that the vertices can be numbered so that the first row of A is $[1\ 1\ 1\ 0\ 0\ 0]$ and the first column is $[1\ 1\ 1\ 0\ 0\ 0]^T.$