How many $4$-regular graphs exist on $8$ vertices?
I found that such a graph can't be disconnectd since if so, then graph can be written as disjoint union of atleast two graphs. $4$ regularity condition says that in any component no. of vertices is atleast $5$, so such graph is not possible.
So all graphs will be connected. One such is $K_{4,4}$. How to find rest?
Further how to find those which alongwith above conditions have even Laplacian eigenvalues.