The differential on operator S(Γ)

Consider a simple graph Gamma = (V(Gamma), E(Gamma)) with n vertices and m edges. Let P be a subset of V(Gamma) and B(P) the set of neighbors of P in V(Gamma)\P. In the study of graphs, the concept of differential refers to a measure of how much the number of edges leaving a set of vertices exceeds the size of that set. Specifically, given a subset P of vertices, the differential of P, denoted by a(P), is defined as | B(P)| - |P|. The differential of Gamma, denoted by a(Gamma), is then defined as the maximum differential over all possible subsets of V(Gamma). Additionally, the subdivision operator S(Gamma) is defined as the graph obtained from Gamma by inserting a new vertex on each edge of Gamma. In this paper, we present results for the differential of graphs on the subdivision operator S(Gamma) where some of these show exact values of a(S(Gamma)) if Gamma belongs to a classical family of graphs. We obtain bounds for a(S(Gamma)) involving invariants of a graph such as order n, size m and maximum degree increment , and we study the realizability of the graph Gamma for any [ ] value of a(S(Gamma)) in the interval n - 2, n(n-1) 2- n + 2 . Moreover, we give a characterization for a(S(Gamma)) using the notion of edge star packing.