  In this paper, we study some graph theoretical properties of two derivative
Euler Phi function set-graphs. For the Euler Phi function $\phi(n)$, $n\in
\mathbb{N}$, the set $S_\phi(n) =\{i:\gcd(i,n)=1, 1\leq i \leq n\}$ and the
vertex set is $\{v_i:i\in S_\phi(n)\}$. Two graphs $G_d(S_\phi(n))$ and
$G_p(S_\phi(n))$, defined with respect to divisibility adjacency and relatively
prime adjacency conditions, are studied.
