Generalized non-coprime graphs of groups

TL;DR

This paper investigates the properties of generalized non-coprime graphs derived from finite groups, especially cyclic and nilpotent groups, identifying conditions for various graph classes and exploring their connections to group theory.

Contribution

It provides necessary and sufficient conditions for these graphs to belong to specific classes and extends the analysis to all finite nilpotent groups, linking to Gruenberg--Kegel graphs.

Findings

Characterization of when the graph is a star, path, cycle, etc.

Widening to all finite nilpotent groups yields no new graphs.

Example of EPPO groups showing contrasting behavior.

Abstract

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.