On the differential central operator C(G)

Abstract. Let G = (V (G),E(G)) be a simple, finite graph. The central operator C(G) of G is obtained by subdividing each edge of G exactly once and adding an edge between every pair of nonadjacent vertices of G. Let D ⊆ V (G), and let B(D) denote the set of neighbors of D in V (G) \ D. The differential ∂(D) of D is defined as ∂(D):= |B(D)| − |D|. The maximum value of ∂(D) over all subsets D ⊆ V (G) is the differential of G, denoted by ∂(G). A set D is called a differential set of G if ∂(D) = ∂(G). In this paper, we establish bounds for ∂(C(G)) in terms of several classical graph invariants and investigate structural properties of differential sets in C(G). Furthermore, we determine the exact value of ∂(C(G)) for certain well-known families of graphs.