On g-Noncommuting Graph of a Finite Group Relative to Its Subgroups

Let H be a subgroup of finite non-abelian group G and g∈G. Z(H,G)={x∈H:xy=yx,∀y∈G}. We introduce the graph ΔH,Gg whose vertex set is G\Z(H,G) two distinct vertices x y are adjacent if x∈H or y∈H [x,y]≠g,g−1, where [x,y]=x−1y−1xy. In this paper, we determine whether tree among other results. also discuss about its diameter connectivity with special attention to dihedral groups.