On the multiplicative Zagreb coindex of graphs

Abstract

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 \(M_1(G) = \sum_{v \in V(G)} d_G(v)^2\) and \(M_2(G) = \sum_{uv \in E(G)} d_G(u)d_G(v)\), respectively, where \(d_G(v)\) is the degree of vertex \(v\) in \(G\). The alternative expression of \(M_1(G)\) is \(\sum_{uv \in E(G)}(d_G(u) + d_G(v))\). Recently Ashrafi, Doslic and Hamzeh introduced two related graphical invariants \(\overline{M}_1(G) = \sum_{uv \notin E(G)}(d_G(u)+d_G(v))\) and \(\overline{M}_2(G) = \sum_{uv \notin E(G)} d_G(u)d_G(v)\) named as first Zagreb coindex and second Zagreb coindex, respectively. Here we define two new graphical invariants \(\overline{\Pi}_1(G) = \prod_{uv \notin E(G)}(d_G(u)+d_G(v))\) and \(\overline{\Pi}_2(G) = \prod_{uv \notin E(G)} d_G(u)d_G(v)\) as the respective multiplicative versions of \(\overline{M}_i\) 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.