Volume 11, Special Issue Dedicated to Professor George A. Grätzer, July 2019, 131-148. Intersection graphs associated with semigroup acts Abdolhossein Delfan, Hamid Rasouli ∗ , and Abolfazl Tehranian This article is dedicated to George A. Grätzer Abstract. The intersection graph (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 non- empty intersection. In this paper, we study some graph-theoretic properties of (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 (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. 1 Introduction and preliminaries In recent decades, assigning graphs to algebraic structures has opened a new direction to study algebraic properties via graph-theoretic properties and * Corresponding author Keywords : S-act, intersection graph, chromatic number, clique number, weakly perfect graph. Mathematics Subject Classification[2010]: 20M30, 05C15, 05C25, 16P70. Received: 4 May 2018, Accepted: 26 July 2018. ISSN: Print 2345-5853, Online 2345-5861. © Shahid Beheshti University 131