  Let G be a finite group with identity e and H \neq \{e\} be a subgroup of G.
The generalized non-coprime graph GAmma_{G,H} of G with respect to H is the
simple undirected graph with G - \{e \}\) as the vertex set and two distinct
vertices a and b are adjacent if and only if \gcd(|a|,|b|) \neq 1 and either a
\in H or b \in H, where |a| is the order of a\in G. In this paper, we study
certain graph theoretical properties of generalized non-coprime graphs of
finite groups, concentrating on cyclic groups. More specifically, we obtain
necessary and sufficient conditions for the generalized non-coprime graph of a
cyclic group to be in the class of stars, paths, cycles, triangle-free,
complete bipartite, complete, unicycle, split, claw-free, chordal or perfect
graphs. Then we show that widening the class of groups to all finite nilpotent
groups gives us no new graphs, but we give as an example of contrasting
behaviour the class of EPPO groups (those in which all elements have prime
power order). We conclude with a connection to the Gruenberg--Kegel graph.
