  The non-solvable graph of a finite group G is a simple graph whose vertices
are the elements of G and there is an edge between x and y if and only if the
subgroup generated by x and y is not solvable. The isolated vertices in the
non-solvable graph are exactly the elements of the solvable radical of G. Given
a finite group G and element x of G, the solvabilizer of x with respect to G is
the set of all elements y of G such that the subgroup generated by x and y is
solvable. The purpose of this paper is to study some properties of the
non-solvable graph and the structure of the solvabilizer of x with respect to G
for every element x of G.
