Abstract
Consider a finite group G which acts on itself such that the relation similar to on G, which is defined by x similar to y iff x is a fixed point of y under this action, is both reflexive and symmetric. Let fix(G) represent the set of fixed points of G, i.e., fix(G)={x is an element of G | xg=x for all g is an element of G}. We associate with G a new graph using G\fix(G) as its set of vertices, connecting vertices x,y is an element of G\fix(G) iff x is a fixed point of y. This graph is referred to as the stabilizing graph of G and is denoted by Delta s(G). Additionally, the complement of the stabilizing graph Delta s(G), denoted by del s(G), is named the nonstabilizing graph of G. The aim of this study is to analyze various properties of the nonstabilizing graph del s(G) and to explore how this graph-theoretical concept relates to the broader understanding of group actions.