On the structure of finite groups associated to regular non-centralizer graphs

The non-centralizer graph of a finite group G is the simple graph TG whose vertices are the elements of G with two vertices are adjacent if their centralizers are distinct. The induced non -centralizer graph of G is the induced subgraph of TG on G \ Z(G). A finite group is called regular (resp. induced regular) if its non-centralizer graph (resp. induced non-centralizer graph) is regular. In this paper we study the structure of regular groups and induced regular groups. We prove that if a group G is regular, then G/Z(G) as an elementary 2-group. Using the concept of maximal centralizers, we succeeded in proving that if G is induced regular, then G/Z(G) is a p-group. We also show that a group G is induced regular if and only if it is the direct product of an induced regular p-group and an abelian group.