Some results on the main supergraph of finite groups

Asboei, Alireza Khalili; Salehi, Syyed Sadegh

doi:10.12958/adm584

Cited by 3 publications

(1 citation statement)

References 14 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Hamiltonianity and Eulerianness of S (G) were investigated in [12] and spectrum and L-spectrum of S (G) were investigated in [11]. In [3], Asboei and Salehi studied the well-known Thompson's problem and recognized the projective special linear groups and the projective linear groups by their order graphs. Recently, Asboei and Salehi [4] identified many families of finite non-solvable groups by their order graphs, which is an important work for the Thompson's problem.…”

Section: Introductionmentioning

confidence: 99%

On connected components and perfect codes of proper order graphs of finite groups

Li,

Lin,

2024

PEAS

View full text Add to dashboard Cite

Let G be a finite group with the identity element e. The proper order graph of G, denoted by S * (G), is an undirected graph with a vertex set G \ {e}, where two distinct vertices x and y are adjacent whenever o(x) | o(y) or o(y) | o(x), where o(x)and o(y) are the orders of x and y, respectively. This paper studies the perfect codes of S * (G). We characterize all connected components of a proper order graph and give a necessary and sufficient condition for a connected proper order graph. We also determine the perfect codes of the proper order graphs of a few classes of finite groups, including nilpotent groups, CP-groups, dihedral groups and generalized quaternion groups.

show abstract

Section: Introductionmentioning

confidence: 99%

On connected components and perfect codes of proper order graphs of finite groups

Li,

Lin,

2024

PEAS

View full text Add to dashboard Cite

show abstract

Graphs on groups in terms of the order of elements: A review

Madhumitha,

Naduvath

2023

Discrete Math. Algorithm. Appl.

View full text Add to dashboard Cite

Two mathematical fields that concentrate on creating and analyzing structures are algebra and graph theory. There are numerous studies linking algebraic structures like groups, rings, fields and vector spaces with graph theory. Several algebraic graphs have been defined based on the properties of the order of the group and its elements. In this paper, we systematically review the literature on such graphs to understand the research dynamics in the field.

show abstract

On forbidden subgraphs of main supergraphs of groups

Xu,

Chen

et al. 2024

era

View full text Add to dashboard Cite

<p>In this study, we explore the main supergraph $ \mathcal{S}(G) $ of a finite group $ G $, defined as an undirected, simple graph with a vertex set $ G $ in which two distinct vertices, $ a $ and $ b $, are adjacent in $ \mathcal{S}(G) $ if the order of one is a divisor of the order of the other. This is denoted as either $ o(a)\mid o(b) $ or $ o(b)\mid o(a) $, where $ o(\cdot) $ is the order of an element. We classify finite groups for which the main supergraph is either a split graph or a threshold graph. Additionally, we characterize finite groups whose main supergraph is a cograph. Our classification extends to finite groups $ G $ with $ \mathcal{S}(G) $, a cograph that includes when $ G $ is a direct product of two non-trivial groups, as well as when $ G $ is either a dihedral group, a generalized quaternion group, a symmetric group, an alternating group, or a sporadic simple group.</p>

show abstract

Some results on the main supergraph of finite groups

Cited by 3 publications

References 14 publications

On connected components and perfect codes of proper order graphs of finite groups

On connected components and perfect codes of proper order graphs of finite groups

Graphs on groups in terms of the order of elements: A review

On forbidden subgraphs of main supergraphs of groups

Contact Info

Product

Resources

About