Multiplicative Zagreb indices of k-trees

Let G be a graph with vertex set V ( G ) and edge set E ( G ) . The first generalized multiplicative Zagreb index of G is ? 1 , c ( G ) = ? v ? V ( G ) d ( v ) c , for a real number c 0 , and the second multiplicative Zagreb index is ? 2 ( G ) = ? u v ? E ( G ) d ( u ) d ( v ) , where d ( u ) , d ( v ) are the degrees of the vertices of u , 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 , 2 , respectively, and extend the results of Gutman to the generalized tree, the k -tree, where the results of Gutman are for k = 1 . Additionally, we characterize the extremal graphs and determine the exact bounds of these indices of k -trees, which attain the lower and upper bounds.