If G=(VG,EG) is a graph of order n, we call S⊆VG an isolating set if the induced by VG−NG[S] contains no edges. The minimum cardinality G called isolation number G, and it denoted ι(G). It known that ι(G)≤n3 bound sharp. A subset dominating in NG[S]=VG. domination number, γ(G). In this paper, analyze family trees T where ι(T)=γ(T), prove ι(T)=n3 implies ι(T)=γ(T). Moreover, give different equiv...

‎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...