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I'm new to graph theory and was wondering:

Can two graphs with every vertex's order being more than 2 and have the same number of vertices and edges be non-isomorphic?

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Yeah it's possible. For example take $n=6$ and each of the vertices have degree $3$. Then you can construct $K_{3,3}$ the complete bipartite graph on $6$ vertices and also another graph which isn't bipartite. The second graph can be obtained by constructing two distinct $C_3$ cycles and then connecting three pairs of vertices from distinct cycles. As both don't share a property (namely bipartivity) then they can't be isomorphic to each other.

You can find a picture of the two non-isomoprhic graphs here.

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    $\begingroup$ That other graph could be visualised as the edges of a triangular prism. $\endgroup$ – Jaap Scherphuis Nov 22 '17 at 13:28

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