On the differential in central operator 𝒞(G)

¹ Unidad Académica de Matemáticas, Universidad Autónoma de Zacatecas "Francisco García Salinas", Paseo La Bufa, Int. Calzada Solidaridad, 98060 Zacatecas, México
² Facultad de Matemáticas, Universidad Autonoma de Guerrero, Carlos E. Adame No. 54, Col. La Garita, 39650 Acapulco, Gro., México

^* Corresponding author: This email address is being protected from spambots. You need JavaScript enabled to view it.

Received: 21 May 2025
Accepted: 22 March 2026

Abstract

The central operator 𝒞(G) is a structural transformation that combines edge subdivision with the complementation on the original vertex set. In this paper, we characterize the behavior of the differential ∂(𝒞(G)), an invariant that measures the maximum influence potential of a network. We establish sharp bounds for ∂(𝒞(G)) in terms of the order n and the maximum degree Δ(G) of the base graph, proving a guaranteed growth property: for any connected graph of order n ≥ 4, ∂(𝒞(G)) ≥ ∂(Ḡ)+1. Our structural analysis reveals a parsimony property of optimal sets. Specifically, Theorem 3.16 shows that minimum differential sets in 𝒞(G) are primarily supported on the original vertex set V, effectively reducing the search space for optimization algorithms. Furthermore, in Theorem 3.22 we establish a notable theoretical convergence by identifying conditions under which the central, subdivision, and ℛ(G) operators yield the same differential value, namely m + n − 4. Finally, we provide exact evaluations for several fundamental graph families and derive Nordhaus-Gaddum type inequalities (Prop. 3.24 and Cor. 3.25) for the class of trees. These results clarify how the central operator enhances the diffusion capacity of networks, thereby bridging the gap between topological transformations and practical influence maximization.