Generalized non-coprime graphs of groups

Let G be a finite group with identity e and \( H \ne \{e\}\) $H\ne \{e\}$ be a subgroup of G. The generalized non-coprime graph \(\varGamma _{G,H}\) ${\Gamma }_{G,H}$ of \( G \) $G$ with respect to \(H\) $H$ is the simple undirected graph with \(G \setminus \{e \}\) $G\backslash \{e\}$ as the vertex set and two distinct vertices \( x \) $x$ and \( y\) $y$ are adjacent if and only if \(\gcd (|x|,|y|) \ne 1\) $gcd(|x|,|y|)\ne 1$ and either \(x \in H\) $x\in H$ or \(y \in H\) $y\in H$ , where |x| is the order of \(x\in G\) $x\in G$ . 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, triangle-free, complete bipartite, complete, 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.