Given a subgroup-closed class of finite groups C and a finite group G, we can define a graph whose vertices are the elements of G and where two vertices x and y are joined by an edge if the subgroup generated by x and y belongs to C. We focus on the case where X is the class of solvable groups and call it the solubility graph. Properties of this graph, in particular the arithmetic and structural properties of vertex neighborhoods can have dramatic consequences on the structure of G. For instance, if a non-identity element has degree p−1 or p2−1 where p is prime, then G is a p-group. We call the neighborhood of vertex x in the solubility graph the solubilizer of x. In this talk, we explore the properties of the solubility graph focusing our attention on solubilizers and discuss how restrictions on the structure of solubilizers can affect the structure of G.