We associate a graph G 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 G and its induced subgraph on G\nilG, where nilG = x ∈ Gx y is nilpotent for all y ∈ G. For any finite group G, we prove that G has either Z∗G or Z∗G+ 1 connected components, where Z∗G is the hypercenter of G. We give a new characterization for finite nilpotent groups in terms of the non- nilpotent graph. In fact, we prove that a finite group G is nilpotent if and only if the set of vertex degrees of G has at most two elements.
