Common Neighborhood Energy of Commuting Graphs of Finite Groups

Nath, Rajat Kanti; Fasfous, Walaa Nabil Taha; Das, Kinkar Ch.; Shang, Yilun

doi:10.3390/sym13091651

Cited by 12 publications

(7 citation statements)

References 26 publications

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“…For future work, the application of this graph to the study on Laplacian eigenvalues of an ideal-based dot total graph, which is closely related to the work in the paper [6], can be investigated. Additionally, the energy of an ideal-based dot total graph, which is related to the recent work in [5,7], requires more consideration. The…”

Section: Discussionmentioning

confidence: 99%

“…Furthermore, if I = (0) in T I (Γ(R)), then T I (Γ(R)) = Γ 0 (R); this graph is studied by Beck [4], in which he considered R as a simple graph for which its vertex set is the set of all elements of R and edge set such that for all distinct x, y ∈ R, e = xy ∈ E(Γ 0 (R)) if and only if xy = 0. In addition, some fundamentals of Laplacian eigenvalues and energy of graphs can be identified in [5][6][7].…”

Section: Introductionmentioning

confidence: 99%

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An Ideal-Based Dot Total Graph of a Commutative Ring

et al. 2021

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In this paper, we introduce and investigate an ideal-based dot total graph of commutative ring R with nonzero unity. We show that this graph is connected and has a small diameter of at most two. Furthermore, its vertex set is divided into three disjoint subsets of R. After that, connectivity, clique number, and girth have also been studied. Finally, we determine the cases when it is Eulerian, Hamiltonian, and contains a Eulerian trail.

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Section: Discussionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

An Ideal-Based Dot Total Graph of a Commutative Ring

et al. 2021

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“…Recent results on the commuting graph of generalized dihedral groups can be found in [31] and the references therein. The connectivity and spectral radius of adjacency matrix of commuting graphs were studied in [5], Laplacian and signless Laplacian spectrum of commuting graphs on dihedral groups were investigated in [3], common neighborhood energy of commuting graphs [36]. For other spectral properties of commuting graphs, we refer to [27] and the references therein.…”

Section: Groupsmentioning

confidence: 99%

On normalized distance Laplacian eigenvalues of graphs and applications to graphs defined on groups and rings

RATHER¹,

GANIE²,

AOUCHICHE³

2022

CJM

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The normalized distance Laplacian matrix of a connected graph $ G $, denoted by $ D^{\mathcal{L}}(G) $, is defined by $ D^{\mathcal{L}}(G)=Tr(G)^{-1/2}D^L(G)Tr(G)^{-1/2}, $ where $ D(G) $ is the distance matrix, the $D^{L}(G)$ is the distance Laplacian matrix and $ Tr(G)$ is the diagonal matrix of vertex transmissions of $ G. $ The set of all eigenvalues of $ D^{\mathcal{L}}(G) $ including their multiplicities is the normalized distance Laplacian spectrum or $ D^{\mathcal{L}} $-spectrum of $G$. In this paper, we find the $ D^{\mathcal{L}} $-spectrum of the joined union of regular graphs in terms of the adjacency spectrum and the spectrum of an auxiliary matrix. As applications, we determine the $ D^{\mathcal{L}} $-spectrum of the graphs associated with algebraic structures. In particular, we find the $ D^{\mathcal{L}} $-spectrum of the power graphs of groups, the $ D^{\mathcal{L}} $-spectrum of the commuting graphs of non-abelian groups and the $ D^{\mathcal{L}} $-spectrum of the zero-divisor graphs of commutative rings. Several open problems are given for further work.

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“…The energy E(G) is the sum of absolute A-eigenvalues of G. A significant amount of research has been conducted on this idea [16]. Following the potential applications of the A-spectrum, numerous topological indices were investigated from a spectral perspective by modifying the classical adjacency matrix accordingly [19][20][21][22][23][24][25][26][27][28][29]. Zhou and Trinajstić [30] introduced the matrix corresponding to the sum-connectivity index and studied associated energy in 2010.…”

Section: Introductionmentioning

confidence: 99%

On Neighborhood Inverse Sum Indeg Energy of Molecular Graphs

et al. 2022

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The spectral graph theory explores connections between combinatorial features of graphs and algebraic properties of associated matrices. The neighborhood inverse sum indeg (NI) index was recently proposed and explored to be a significant molecular descriptor. Our aim is to investigate the NI index from a spectral standpoint, for which a suitable matrix is proposed. The matrix is symmetric since it is generated from the edge connection information of undirected graphs. A novel graph energy is introduced based on the eigenvalues of that matrix. The usefulness of the energy as a molecular structural descriptor is analyzed by investigating predictive potential and isomer discrimination ability. Fundamental mathematical properties of the present spectrum and energy are investigated. The spectrum of the bipartite class of graphs is identified to be symmetric about the origin of the real line. Bounds of the spectral radius and the energy are explained by identifying the respective extremal graphs.

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Common Neighborhood Energy of Commuting Graphs of Finite Groups

Cited by 12 publications

References 26 publications

An Ideal-Based Dot Total Graph of a Commutative Ring

An Ideal-Based Dot Total Graph of a Commutative Ring

On normalized distance Laplacian eigenvalues of graphs and applications to graphs defined on groups and rings

On Neighborhood Inverse Sum Indeg Energy of Molecular Graphs

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