Neighborhoods, connectivity, and diameter of the nilpotent graph of a finite group

The nilpotent graph of a group G is the simple and undirected graph whose vertices are the elements of G and two distinct vertices are adjacent if they generate a nilpotent subgroup of G. Here we discuss some topological properties of the nilpotent graph of a finite group G. Indeed, we characterize finite solvable groups whose closed neighborhoods are nilpotent subgroups. Moreover, we study the connectivity of the graph $\Gamma \left(G\right)$ obtained removing all universal vertices from the nilpotent graph of G. Some upper bounds to the diameter of $\Gamma \left(G\right)$ are provided when G belongs to some classes of groups.