2022/08/03 by S. Anukumar Kathirvel, Peter J. Cameron, Kathirvel, S. Anukumar +2
Chemistry · Neuroscience · #05C25 #Click Chemistry and Applications #Combinatorics (math.CO) #FOS: Mathematics #Group Theory (math.GR) #Nuclear Receptors and Signaling
paper · pdf · doi:10.48550/arxiv.2208.01900
openalex publication_date 2022/08/03 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Let G be a finite group with identity e and H ≠ \e\ be a subgroup of G. The generalized non-coprime graph GAmmaG,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|) ≠ 1 and either a ∈ H or b ∈ H, where |a| is the order of a∈ 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.