the narumi-katayama index was the first topological index defined by the product of some graph theoretical quantities. let $g$ be a simple graph with vertex set $v = {v_1,ldots, v_n }$ and $d(v)$ be the degree of vertex $v$ in the graph $g$. the narumi-katayama index is defined as $nk(g) = prod_{vin v}d(v)$. in this paper, the narumi-katayama index is generalized using a $n$-ve...