On the intersection graph of a finite group

Shahsavari, Hossein; Khosravi, Behrooz

doi:10.21136/cmj.2017.0446-16

Cited by 10 publications

(8 citation statements)

References 12 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…The number of edges in a graph that intersect a vertex, or degree, is shown by the symbol 𝑑 𝐺 (𝑥) [17]. Any graph where 𝑑 Γ (𝑥) = 𝑑 Γ for any 𝑥, 𝑦 ∈ Γ(𝑉) is said to be regular [7] . If two unique vertex pairs x and y exist in a graph, the path between them is defined as a sequence of vertex pairs 𝑥 = 𝑦 0 , 𝑦 1 , … , 𝑦 𝑛 = 𝑦 such that(𝑦 𝑖 , 𝑦 𝑖+1 ) ∈ 𝐸(Γ) for 0 ≤ 𝑖 ≤ 𝑛 − 1, and n is referred to as the length of this path [7] .…”

Section: Preliminariesmentioning

confidence: 99%

“…On the Cayley graph, many intriguing outcomes have been found see [2][3][4][5][6]. The intersection graph of a finite group G, denoted as 𝛽(G), is an undirected graph whose vertices are all nontrivial proper subgroups of G and whose two different vertices H and K are close when H ∩ K≠ 1 [7] . The independence graph of a finite groups is a graph whose vertices are the components G and where two vertices, x and y, are adjacent if there is a minimal generating set of G that includes x and y [8].…”

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

p-Graphs Associated with Some Groups and Vice Versa

Hatoo

2022

basjs

View full text Add to dashboard Cite

show abstract

Section: Preliminariesmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

p-Graphs Associated with Some Groups and Vice Versa

Hatoo

2022

basjs

View full text Add to dashboard Cite

show abstract

“…Tamizh et al [31], continued the seminal paper of Csákány and Pollák to introduce the subgroup intersection graph of a finite group G. Further, in [25], it was shown that the diameter of intersection graph of a finite non-abelian simple group has an upper bound 28. Shahsavari et al [29] have studied the structure of the automorphism group of this graph. The intersection graph on cyclic subgroups of a group has been studied in [14].…”

Section: Introductionmentioning

confidence: 99%

On the intersection ideal graph of semigroups

Baloda¹,

Kumar²

2022

Preprint

View full text Add to dashboard Cite

The intersection ideal graph Γ(S) of a semigroup S is a simple undirected graph whose vertices are all nontrivial left ideals of S and two distinct left ideals I, J are adjacent if and only if their intersection is nontrivial.In this paper, we investigate the connectedness of Γ(S). We show that if Γ(S) is connected then diam(Γ(S)) ≤ 2.Further we classify the semigroups such that the diameter of their intersection graph is two. Other graph invariants, namely perfectness, planarity, girth, dominance number, clique number, independence number etc. are also discussed.Finally, if S is union of n minimal left ideals then we obtain the automorphism group of Γ(S).Recently, Chakrabarty et al. [8] introduced the notion of intersection ideal graph of rings. The intersection ideal graph Γ(R) of a ring R is an undirected simple graph whose vertex set is the collection of nontrivial left ideals of R and two distinct vertices I, J are adjacent if and only if I ∩ J = {0}. They characterized the rings R for which the graph Γ(R) is connected and obtain several necessary and sufficient conditions on a ring R such that Γ(R) is complete. Planarity of intersection graphs of ideals of ring with unity is described in [15] and domination number in 2010 Mathematics Subject Classification. 05C25.

show abstract

“…Ma [12] reduced this upper bound to 28 in 2016. In the other direction, Shasavari and Khosravi [13,Theorem 3.7] proved in 2017 that diam(∆ G ) 3.…”

Section: Introductionmentioning

confidence: 99%

The intersection graph of a finite simple group has diameter at most 5

Freedman

2020

Preprint

View full text Add to dashboard Cite

Let G be a non-abelian finite simple group. In addition, let ∆G be the intersection graph of G, whose vertices are the proper nontrivial subgroups of G, with distinct subgroups joined by an edge if and only if they intersect nontrivially. We prove that the diameter of ∆G has a tight upper bound of 5, thereby resolving a question posed by Shen (2010).Furthermore, a diameter of 5 is achieved only by the baby monster group and certain unitary groups of odd prime dimension.

show abstract

On the intersection graph of a finite group

Abstract: Institute of Mathematics of the Czech Academy of Sciences provides access to digitized documents strictly for personal use. Each copy of any part of this document must contain these Terms of use.

Cited by 10 publications

References 12 publications

p-Graphs Associated with Some Groups and Vice Versa

p-Graphs Associated with Some Groups and Vice Versa

On the intersection ideal graph of semigroups

The intersection graph of a finite simple group has diameter at most 5

Contact Info

Product

Resources

About