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

Monalisha Sharma, Rajat Kanti Nath and Yilun Shang
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Monalisha Sharma: Department of Mathematical Sciences, Tezpur University, Tezpur 784028, India
Rajat Kanti Nath: Department of Mathematical Sciences, Tezpur University, Tezpur 784028, India
Yilun Shang: Department of Computer and Information Sciences, Northumbria University, Newcastle NE1 8ST, UK

Mathematics, 2021, vol. 9, issue 23, 1-13

Abstract: Let H be a subgroup of a finite non-abelian group G and g ∈ G . Let Z ( H , G ) = { x ∈ H : x y = y x , ∀ y ∈ G } . We introduce the graph Δ H , G 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 − 1 y − 1 x y . In this paper, we determine whether Δ H , G g is a tree among other results. We also discuss about its diameter and connectivity with special attention to the dihedral groups.

Keywords: finite group; g -noncommuting graph; connected graph (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2021
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