We associate a graph TG with a group G (called the non-nilpotent graph of G) as follows: take G as the vertex set and two vertices are adjacent if they generate a non-nilpotent subgroup. In this article, we study the graph theoretical properties of TG and its induced subgraph on G\nilG. For any finite group G, we prove that TG has either ZG or Z+ 1 connected components.
