Regularity of extended conjugate graphs of finite groups

The extended conjugate graph associated to a finite group $G$ is defined as an undirected graph with vertex set $G$ such that two distinct vertices joined by an edge if they are conjugate. In this article, we show that several properties of finite groups can be expressed in terms of properties of their extended conjugate graphs. In particular, we show that there is a strong connection between a graph-theoretic property, namely regularity, and an algebraic property, namely nilpotency. We then give some sufficient conditions and necessary conditions for the non-central part of an extended conjugate graph to be regular. Finally, we study extended conjugate graphs associated to groups of order $pq$, $p^3$, and $p^4$, where $p$ and $q$ are distinct primes.