On ‎c‎omputing the general Narumi-Katayama index of some ‎graphs

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 $NK(G) = prod_{vin V}d(v)$‎. ‎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.