In this paper, we define the inclusion graph Inc(A) of an S-act A which is a graph whose vertices are non-trivial subacts of A and two distinct vertices B 1 , B 2 are adjacent if B 1 ⊂ B 2 or B 2 ⊂ B 1 . We investigate the relationship between the algebraic properties of an S-act A and the properties of the graph Inc(A). Some properties of Inc(A) including girth, diameter and connectivity are studied. We characterize some classes of graphs which are the inclusion graphs of S-acts. Finally, some results concerning the domination number of such graphs are given.

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The intersection graph Int(A) of an S-act A over a semigroup S is an undirected simple graph whose vertices are non-trivial subacts of A, and two distinct vertices are adjacent if and only if they have a nonempty intersection. In this paper, we study some graph-theoretic properties of Int(A) in connection to some algebraic properties of A. It is proved that the finiteness of each of the clique number, the chromatic number, and the degree of some or all vertices in Int(A) is equivalent to the finiteness of the number of subacts of A. Finally, we determine the clique number of the graphs of certain classes of S-acts.