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

Sharma, Monalisha, Nath, Rajat Kanti and Shang, Yilun (2021) On g-Noncommuting Graph of a Finite Group Relative to Its Subgroups. Mathematics, 9 (23). p. 3147. ISSN 2227-7390

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Abstract

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

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