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

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