An Ideal-Based Dot Total Graph of a Commutative Ring

Ashraf, Mohammad; Asalool, Jaber H.; Alanazi, Abdulaziz M.; Alamer, Ahmed

doi:10.3390/math9233072

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2023

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Cited by 2 publications

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“…Much the same direction is followed in the second paper [22], where a t-graph associated with a finitely generated group, using the Minkowski metric, is defined. The groups involved here are the two-generator finite groups, and the authors characterize the chromatic number of a t-graph depending exclusively on the parity of t. Finally, the study presented in the eighteenth paper [23] of this collection is focused on the construction and properties of an ideal-based dot total graph associated with a commutative ring with nonzero unity.…”

Section: Contributionsmentioning

confidence: 99%

Preface to the Special Issue “Algebraic Structures and Graph Theory”

Cristea,

Bordbar

2023

Mathematics

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Section: Contributionsmentioning

confidence: 99%

Preface to the Special Issue “Algebraic Structures and Graph Theory”

Cristea,

Bordbar

2023

Mathematics

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$ A_{\alpha} $ matrix of commuting graphs of non-abelian groups

Rather¹,

Ali

Ullah

et al. 2022

MATH

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<abstract><p>For a finite group $ \mathcal{G} $ and a subset $ X\neq \emptyset $ of $ \mathcal{G} $, the commuting graph, indicated by $ G = \mathcal{C}(\mathcal{G}, X) $, is the simple connected graph with vertex set $ X $ and two distinct vertices $ x $ and $ y $ are edge connected in $ G $ if and only if they commute in $ X $. The $ A_{\alpha} $ matrix of $ G $ is specified as $ A_{\alpha}(G) = \alpha D(G)+(1-\alpha) A(G), \; \alpha\in[0, 1] $, where $ D(G) $ is the diagonal matrix of vertex degrees while $ A(G) $ is the adjacency matrix of $ G. $ In this article, we investigate the $ A_{\alpha} $ matrix for commuting graphs of finite groups and we also find the $ A_{\alpha} $ eigenvalues of the dihedral, the semidihedral and the dicyclic groups. We determine the upper bounds for the largest $ A_{\alpha} $ eigenvalue for these graphs. Consequently, we get the adjacency eigenvalues, the Laplacian eigenvalues, and the signless Laplacian eigenvalues of these graphs for particular values of $ \alpha $. Further, we show that these graphs are Laplacian integral.</p></abstract>

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An Ideal-Based Dot Total Graph of a Commutative Ring

Cited by 2 publications

References 14 publications

Preface to the Special Issue “Algebraic Structures and Graph Theory”

Preface to the Special Issue “Algebraic Structures and Graph Theory”

$ A_{\alpha} $ matrix of commuting graphs of non-abelian groups

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