Trees, Unicyclic, and Bicyclic Graphs Extremal with Respect to Multiplicative Sum Zagreb Index∗

For a (molecular) graph G with vertex set V (G) and edge set E(G), the first Zagreb index of G is defined as M1(G) = ∑ v∈V (G) dG(v) 2 where dG(v) is the degree of vertex v in G. The alternative expression for M1(G) is ∑ uv∈E(G)(dG(u)+dG(v)). Very recently, Eliasi, Iranmanesh and Gutman [7] introduced a new graphical invariant ∏∗ 1(G) = ∏ uv∈E(G)(dG(u) + dG(v)) as the multiplicative version of M1. Here we call this new index the multiplicative sum Zagreb index. We characterize the trees, unicylcic, and bicyclic graphs extremal (maximal and minimal) with respect to the multiplicative sum Zagreb index. Moreover, we use a method different but shorter than that in [7] for determining the minimal multiplicative sum Zagreb index of trees.