  The non-centralizer graph of a finite group $G$ is the simple graph
$\Upsilon_G$ whose vertices are the elements of $G$ with two vertices $x$ and
$y$ are adjacent if their centralizers are distinct. The induced subgroup of
$\Upsilon_G$ associated with the vertex set $G\setminus Z(G)$ is called the
induced non-centralizer graph of $G$. The notions of non-centralizer and
induced non-centralizer graphs were introduced by Tolue in \cite{to15}. 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 as well as induced regular groups. Among the
many obtained results, we prove that if a group $G$ is regular (resp. induced
regular) then $G/Z(G)$ as an elementary $2-$group (resp. $p-$group).
