On Multiplicative and Redefined Version of Zagreb Indices of V-Phenylenic Nanotubes and Nanotorus

Let Gbean arbitrary simpleandconnected graph, with the vertex set V(G)and edge set E(G). The first Zagreb index of a graph G is defined as ()1M G=2( )vv V Gd∈∑, where duand dvare the degrees of u and v,respectively. An alternative expression for ()1M G is()( )uve uv E Gdd= ∈+∑. And similarly, the Second Zagreb index ()2M G= ()( )uve uv E Gdd= ∈×∑. In this paper, we consider a multiplicative version of ()1M Gand ()2M Gand define them as ()1PM G=()( )e uE Gvvvdd= ∈+∏and PM G=()( )e uE Gvvvdd= ∈×∏, respectively. And then we compute these Multiple Zagreb indices of V-Phenylenic Nanotubes VPHX[p,q] and V-Phenylenic Nanotorus VPHY[p,q] (∀p,q∈ℕ-{1}). Furthermore, we present the redefined Zagreb indices of V-Phenylenic Nanotubes VPHX[p,q] and V-Phenylenic Nanotorus VPHY[p,q] (∀p,q∈ℕ-{1}).