Action graph of a semigroup act & its functorial connection

In this paper we define C-induced action graph G(S,a,C;A) corresponding to a semigroup act (S,a,A) and subset C of S. This generalizes many interesting graphs including Cayley Graph groups semigroups, Transformation Graphs (TRAG), Group Action (GAG), Derangement Graphs, Directed Power Semigroups etc. We focus on the case when = S name digraph, so obtained, as Semigroup Act (S, a, A). Some basic structural properties follow from algebraic underlying its set. strongly faithful is also studied theoretic characterization well that idempotents in are obtained. introduce notion transitive digraphs based characterize acts class simple digraphs. The fact morphism between leads digraph homomorphism graphs, motivates us represent construction functor category certain capture functorial properties, some which signify previous results terms Category Theory.