The differential on Graph Operator S(G). Gerardo Reyna Faculty of Mathematics, Autonomous University of Guerrero. Carlos E. Adame 5, Col. La Garita, Acapulco, Guerrero, Mexico Jair Castro Simon Faculty of Mathematics, Autonomous University of Guerrero. Carlos E. Adame 5, Col. La Garita, Acapulco, Guerrero, Mexico Omar Rosario Faculty of Mathematics, Autonomous University of Guerrero. Carlos E. Adame 5, Col. La Garita, Acapulco, Guerrero, Mexico Resumen Let G =(V (G),E(G)) be a simple graph with vertex set V (G) and edge set E(G). Let S be a subset of V (G), and let B(S ) be the set of neighbours of S in V (G)\S . The differential ∂ (S ) of S is defined as |B(S )|−|S |. The maximum value of ∂ (S ) taken over all subsets S ⊆ V is the differential ∂ (G) of G. A graph operator is a mapping F : G → G ′ , where G and G ′ are families of graphs. The graph S(G) is defined as the graph obtained from G con bipartici´ on de v´ ertices V (G) ∪ E(G), donde hay tantas aristas entre v ∈ V (G) y e ∈ E(G), como veces e sea incidente con v en G. In this paper we study the relationship between ∂ (G) and ∂ (S(G)). Besides, we relate the differential of a graph with known parameters of a graph, namely, its domination and independence number. Keywords: Differential of a graph; Operators Graphs; Differential. AMS Subject Classification numbers: 05C69; 05C76 Email addresses: gerardoreynah@hotmail.com (Gerardo Reyna), castrosimonjair@gmail.com (Jair Castro Simon), omarrosarioc@gmail.com (Omar Rosario). Preprint submitted to Elsevier 21 de junio de 2021 arXiv:2106.09829v1 [math.CO] 17 Jun 2021