The line graph

of a simple graph

is the graph obtained by taking the edges of

as vertices, and joining two of these vertices whenever the corresponding edges of

have a vertex in common. Given

, it might be impossible to find

; for instance, if

. The complement of a simple graph

is obtained by taking the vertices of

and joining two of them whenever they are not joined in

. Complements of complete graphs are always empty graphs (without edges) and vice versa. The square, cube, or in general, the

power of a graph

is obtained by taking the vertices of

and joining them if there is a path of length at most

joining them. The powers of complete graphs are isomorphic to themselves. Can you find a graph such that its square is different from its cube? Can you find a graph such that its cube is not complete?