We investigate properties which ensure that a given finite graph is the
commuting graph of a group or semigroup. We show that all graphs on at least
two vertices such that no vertex is adjacent to all other vertices is the
commuting graph of some semigroup. Moreover, we obtain a complete
classification of the graphs with an isolated vertex or edge that are the
commuting graph of a group and the cycles that are the commuting graph of a
centrefree semigroup