^{}Department of Mathematics, Eslamshahr Branch, Islamic Azad University, Tehran, Iran.

Abstract

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 $NK(G) = \prod_{v\in V}d(v)$. In this paper, the Narumi-Katayama index is generalized using a $n$-vector $x$ and it is denoted by $GNK(G, x)$ for a graph $G$. Then, we obtain some bounds for $GNK$ index of a graph $G$ by terms of clique number and independent number of $G$. Also we compute the $GNK$ index of splice and link of two graphs. Finally, we use from our results to compute the $GNK$ index of a class of dendrimers.