Degree exponent polynomial of graphs obtained by some graph operations

Ramane, Harishchandra S.; Shinde, Sumedha

doi:10.1016/j.endm.2017.11.010

Cited by 10 publications

(9 citation statements)

References 4 publications

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“…Proposition 2.1. [19] If 𝑤, 𝑥, 𝑦 and 𝑧 are real numbers, and 𝐽 𝑛 is an 𝑛 × 𝑛 matrix whose all elements are equal to 1, then the determinant of the (𝑛 1 + 𝑛 2 ) × (𝑛 1 + 𝑛 2 ) matrix of the form…”

Section: Preliminariesmentioning

confidence: 99%

On the Spectral Radius and Sombor Energy of the Non-Commuting Graph for Dihedral Groups

Romdhini,

Nawawi

2024

Mal. J. Fund. Appl. Sci.

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The non-commuting graph, denoted by , is defined on a finite group , with its vertices are elements of excluding those in the center of . In this graph, two distinct vertices are adjacent whenever they do not commute in . The graph can be associated with several matrices including the most basic matrix, which is the adjacency matrix, , and a matrix called Sombor matrix, denoted by . The entries of are either the square root of the sum of the squares of degrees of two distinct adjacent vertices, or zero otherwise. Consequently, the adjacency and Sombor energies of is the sum of the absolute eigenvalues of the adjacency and Sombor matrices of , respectively, whereas the spectral radius of is the maximum absolute eigenvalues. Throughout this paper, we find the spectral radius obtained from the spectrum of and the Sombor energy of for dihedral groups of order , where . Moreover, there is an almost linear correlation between the Sombor energy and the adjacency energy of for which is slightly different than reported earlier in previous literature.

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Section: Preliminariesmentioning

confidence: 99%

On the Spectral Radius and Sombor Energy of the Non-Commuting Graph for Dihedral Groups

Romdhini,

Nawawi

2024

Mal. J. Fund. Appl. Sci.

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“…Lemma 2.1. [25] If w, x, y and z are real numbers, and I n be the n × n identity matrix and J n be the n × n matrix whose all entries are equal to 1, then the determinant of the (n 1 + n 2 ) × (n 1 + n 2 ) matrix of the form…”

Section: Preliminariesmentioning

confidence: 99%

Neighbors Degree Sum Energy of Commuting and Non-Commuting Graphs for Dihedral Groups

Romdhini¹,

Nawawi²,

Chen³

2023

MJMS

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The neighbors degree sum (NDS) energy of a graph is determined by the sum of its absolute eigenvalues from its corresponding neighbors degree sum matrix. The non-diagonal entries of NDS−matrix are the summation of the degree of two adjacent vertices, or it is zero for non-adjacent vertices, whereas for the diagonal entries are the negative of the square of vertex degree. This study presents the formulas of neighbors degree sum energies of commuting and non-commuting graphs for dihedral groups of order 2n, D2n, for two cases−odd and even n. The results in this paper comply with the well known fact that energy of a graph is neither an odd integer nor a square root of an odd integer.

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“…Lemma 2.1: (Ramane & Shinde, 2017) If 𝒂, 𝒃, 𝒄 and 𝒅 are real numbers and 𝑱 𝒏 is an 𝒏 × 𝒏 matrix whose entries are equal to one, then the determinant of the (𝒏 𝟏 + 𝒏 𝟐 ) × (𝒏 𝟏 + 𝒏 𝟐 ) matrix of the form…”

Section: Preliminariesmentioning

confidence: 99%

Degree Sum Energy of Non-Commuting Graph for Dihedral Groups

Romdhini

Nawawi²

2022

MJS

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For a finite group G, let Z(G) be the centre of G. Then the non-commuting graph on G, denoted by ΓG, has G\Z(G) as its vertex set with two distinct vertices vp and vq joined by an edge whenever vpvq ≠ vqvp. The degree sum matrix of a graph is a square matrix whose (p,q)-th entry is dvp + dvq whenever p is different from q, otherwise, it is zero, where dvi is the degree of the vertex vi. This study presents the general formula for the degree sum energy, EDS (ΓG), for the non-commuting graph of dihedral groups of order 2n, D2n, for all n ≥ 3.

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Degree exponent polynomial of graphs obtained by some graph operations

Cited by 10 publications

References 4 publications

On the Spectral Radius and Sombor Energy of the Non-Commuting Graph for Dihedral Groups

On the Spectral Radius and Sombor Energy of the Non-Commuting Graph for Dihedral Groups

Neighbors Degree Sum Energy of Commuting and Non-Commuting Graphs for Dihedral Groups

Degree Sum Energy of Non-Commuting Graph for Dihedral Groups

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