On the multiplicative zagreb coindex of graphs

For a (molecular) graph G with vertex set V (G) and edge set E(G), the first and second Zagreb indices of G are defined as M1(G) = Pv2V (G) dG(v)2 and M2(G) = Puv2E(G) dG(u)dG(v), respectively, where dG(v) is the degree of vertex v in G. The alternative expression of M1(G) is Puv2E(G)(dG(u) + dG(v)). Recently Ashrafi, Došlić and Hamzeh introduced two related graphical invariants M1(G) = Puv /2E(G)(dG(u)+dG(v)) and M2(G) = Puv/2E(G) dG(u)dG(v) named as first Zagreb coindex and second Zagreb coindex, respectively. Here we define two new graphical invariants 1(G) = Quv/2E(G)(dG(u)+dG(v)) and 2(G) = Quv/2E(G) dG(u)dG(v) as the respective multiplicative versions of Mi for i = 1, 2. In this paper, we have reported some properties, especially upper and lower bounds, for these two graph invariants of connected (molecular) graphs. Moreover, some corresponding extremal graphs have been characterized with respect to these two indices.