A Research About Non-coprime Graph Of Finite Groups

Let G be a finite group.The element non-coprime graph ΓG of G is the graph with vertices the non-identity elements of G,in which two distinct vertices x and y adjacent if and only if(|x|,|y|)≠ 1.This article introduces a new definition,called subgroup non-coprime graph.The graph Γ2G of G is the graph with vertices the non-trivial proper subgroups of G,where two distinct vertices A and B are adjacent if and only if(|A|,|B|)≠ 1.The main content and conclusion of this article:(1)Based on the definition and properties of element non-coprime graphs,First,we give planarization for element non-coprime graph ΓG of G if and only if G≌Zn,Z2×Z2,or S3,and n ≤6;Let G is z finite group,orientable genus γ(ΓG)<1 for element non-coprime graph ΓG of G if and only if G≌Zn,D10,Z2 × Z2 or S3,n is positive integer and n ≤ 7;nonorientable genus γ(ΓG)≤ 1 for element non-coprime graph ΓG of G if and only if G≌Zn,Z2×Z2,Z2×Z2×Z2,Z4×Z2,D14,D10,S3,D8 or Q8,nis positive integer and n ≤ 8.Secondly,we give the computation of chromatic number and clique number of element non-coprime graph ΓD2n.(2)Based on the learning of element non-prime graphs,the definition of subgroup non coprime graphs is given and research some necessity of connectivity for subgroup non-prime graph(Contains upper boundary of diameter,n-regular,etc)and subgroup non-prime graph of planarization,and given the proof process of the conclusion.