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# Transformations on Graphs

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?

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