I want to find another graph that has 6 vertices and each has degree $3$ that is not isomorphic to these two graphs below. I know that these two graphs are isomorphic. They will all have the same degree sequence, So I kind of stuck how to find such graph. Any suggestions ?
3 Answers
$\begingroup$Draw two triangles and join them with a 1-factor.
$\endgroup$ 6 $\begingroup$A graph on $6$ vertices is regular of degree $3$ if and only if its complement is regular of degree $2.$
First find two nonisomorphic $2$-regular graphs on $6$ vertices (hint: one is connected, the other is not); their complements will be nonisomorphic $3$-regular graphs on $6$ vertices.
$\endgroup$ $\begingroup$This graph being $3-regular$ on $6$ vertices always contain exactly $9$ edges.
One possible graph is as follows:
-Join the vertices $1,2,3,4,5,6$ by making parallel edges between each pair $1$ & $2$, $3$ & $4$, and $5$ & $6$. This will exhaust $6$ of your edges. Now join each pair $2$ & $3$, $4$ & $5$, $6$ & $1$ by using $3$ more edges. -
As this graph is not simple hence cannot be isomorphic to any graph you have given.
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