Isomorphic $g$-noncommuting graphs of finite groups

Nasiri, Mahboube; Erfanian, Ahmad; Ganjali, Masoumeh; Jafarzadeh, Abbas

doi:10.5486/pmd.2017.7577

Cited by 3 publications

(8 citation statements)

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“…In this paper, we have obtained results on Γ r R and ∆ r R analogous to certain results for g-noncommuting graphs of finite groups obtained in [18,20]. Of course, we have obtained results not analogous to the results for g-noncommuting graphs of finite groups.…”

Section: (B)supporting

confidence: 49%

“…There are many generalizations of noncommuting graphs of finite groups. The g-noncommuting graph of a finite group, studied extensively in [17][18][19][20], is a kind of generalization of noncommuting graph of a finite group. It is worth mentioning that commuting/noncommuting graphs and their generalizations for finite rings are not much studied.…”

Section: Introductionmentioning

confidence: 99%

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On r-Noncommuting Graph of Finite Rings

Nath

Sharma

Dutta³

et al. 2021

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Let R be a finite ring and r∈R. The r-noncommuting graph of R, denoted by ΓRr, is a simple undirected graph whose vertex set is R and two vertices x and y are adjacent if and only if [x,y]≠r and [x,y]≠−r. In this paper, we obtain expressions for vertex degrees and show that ΓRr is neither a regular graph nor a lollipop graph if R is noncommutative. We characterize finite noncommutative rings such that ΓRr is a tree, in particular a star graph. It is also shown that ΓR1r and ΓR2ψ(r) are isomorphic if R1 and R2 are two isoclinic rings with isoclinism (ϕ,ψ). Further, we consider the induced subgraph ΔRr of ΓRr (induced by the non-central elements of R) and obtain results on clique number and diameter of ΔRr along with certain characterizations of finite noncommutative rings such that ΔRr is n-regular for some positive integer n. As applications of our results, we characterize certain finite noncommutative rings such that their noncommuting graphs are n-regular for n≤6.

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Section: (B)supporting

confidence: 49%

Section: Introductionmentioning

confidence: 99%

On r-Noncommuting Graph of Finite Rings

Nath

Sharma

Dutta³

et al. 2021

Axioms

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“…It is well known that Z(D 2n ) = {1}, the commutator subgroup D 2n = a if n is odd and Z(D 2n ) = {1, a n 2 } and D 2n = a 2 if n is even. By (Theorem 4, [19]), it follows that ∆ g D 2n is disconnected if n = 3, 4, 6. Therefore, we consider n ≥ 8 and n ≥ 5 accordingly, as n is even or odd in the following results.…”

Section: Connectivity and Diametermentioning

confidence: 94%

“…Connectivity of ∆ g G was studied in [19][20][21]. It was conjectured that the diameter of ∆ g G is equal to 2 if ∆ g G is connected.…”

Section: Connectivity and Diametermentioning

confidence: 99%

“…In addition, ∆ g H,G is an induced subgraph of Γ g H,G , studied by the authors in [18], induced by G \ Z(H, G). If H = G and g = 1, then ∆ g H,G := Γ G , the non-commuting graph of G. If H = G, then ∆ g H,G := ∆ g G , a generalization of Γ G called an induced g-noncommuting graph of G on G \ Z(G) studied extensively in [19][20][21] by Erfanian and his collaborators.…”

Section: Introductionmentioning

confidence: 99%

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

2021

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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 ΔH,Gg 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 ΔH,Gg 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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$g$-noncommuting graph of a finite group relative to its subgroups

Sharma¹,

Nath²

2020

Preprint

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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 ∆ g H,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 xy. In this paper, we determine whether ∆ g H,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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Isomorphic $g$-noncommuting graphs of finite groups

Cited by 3 publications

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On r-Noncommuting Graph of Finite Rings

On r-Noncommuting Graph of Finite Rings

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

$g$-noncommuting graph of a finite group relative to its subgroups

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