Multiplicative Zagreb indices of kk-trees

Let GG be a graph with vertex set V(G)V(G) and edge set E(G)E(G). The first generalized multiplicative Zagreb index of GG is ∏1,c(G)=∏v∈V(G)d(v)c∏1,c(G)=∏v∈V(G)d(v)c, for a real number c>0c>0, and the second multiplicative Zagreb index is ∏2(G)=∏uv∈E(G)d(u)d(v)∏2(G)=∏uv∈E(G)d(u)d(v), where d(u),d(v)d(u),d(v) are the degrees of the vertices of u,vu,v. The multiplicative Zagreb indices have been the focus of considerable research in computational chemistry dating back to Narumi and Katayama in 1980s. In this paper, we generalize Narumi–Katayama index and the first multiplicative index, where c=1,2c=1,2, respectively, and extend the results of Gutman to the generalized tree, the kk-tree, where the results of Gutman are for k=1k=1. Additionally, we characterize the extremal graphs and determine the exact bounds of these indices of kk-trees, which attain the lower and upper bounds.