On the difference of Zagreb coindices of graph operation

Abstract

Recently the differences of Zagreb indices are reported by Furtula, Gutman and Ediz. The Zagreb coindices are defined as M₁(G) = ∑_{uv ∉ E(G)} [d(u) + d(v)] and M₂(G) = ∑_{uv ∉ E(G)}d(u)d(v), where d(u) is the degree of the vertex u of G. So far, the difference M₁ and M₂ has not been studied. We show that this difference is closely related to the vertex degree-based invariant RM₂=∑_{uv ∉ E(G)}(d(u)-1)(d(v)-1), and explore their basic mathematical properties and present explicit formulas for this new invariant under several graph operations.