On <i>α</i>-adjacency energy of graphs and Zagreb index

Pirzada, S.; Rather, Bilal A.; Ganie, Hilal A.; Shaban, Rezwan Ul

doi:10.1080/09728600.2021.1917973

Cited by 9 publications

(7 citation statements)

References 21 publications

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“…As A α (G) is a symmetric matrix, for α ∈ 1 2 , 1 , clearly A α (G) is positive semidefinite and so the A α eigenvalues of G can be taken as ρ 1 (G) ≥ ρ 2 (G) ≥ • • • ≥ ρ n (G). In this setup, the matrices A(G), Q(G) and D(G) were seen in a new light and very interesting results were deduced in [3,10,11,14,17].…”

Section: L(g)mentioning

confidence: 76%

“…Lemma 1.3. [10,14] Let G be a connected graph of order n and size m having vertex degree sequence {d 1 , d 2 , . .…”

Section: L(g)mentioning

confidence: 99%

“…Lemma 1.6. [14] Let G be a connected graph of order n and size m, where m ≥ n and let G ′ = G − e be a connected graph obtained from G by deleting an edge. Then…”

Section: L(g)mentioning

confidence: 99%

“…[10] The A α eigenvalues of the complete graph K n are {n − 1, (αn − 1) [n−1] }, where [j] means the multiplicity of λ. Lemma 1.8. [14] Let G be a connected graph of order n having vertex degree sequence…”

Section: L(g)mentioning

confidence: 99%

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On the sum of the powers of $ A_\alpha $ eigenvalues of graphs and $ A_\alpha $-energy like invariant

Pirzada

Rather

Shaban

et al. 2022

bspm

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For a connected simple graph $ G $ with $ A_{\alpha} $ eigenvalues $ \rho_{1}\geq\rho_{2}\geq\dots\geq\rho_{n} $ and a real number $\beta $, let $ S_{\beta}^{\alpha}(G) =\sum\limits_{i=1}^{n}\rho_{i}^{\beta}$ be the sum of the $ \beta^{th} $ powers of the $ A_{\alpha} $ eigenvalues of graph $ G $. In this paper, we obtain various bounds for the graph invariant $ S_{\beta}^{\alpha}(G) $ in terms of different graph parameters. As a consequence, we obtain the bounds for the quantity $ IE^{A_{\alpha}}(G)= S_{\frac{1}{2}}^{\alpha}(G),$ the $ A_{\alpha} $ energy-like invariant of the graph $ G .$

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Section: L(g)mentioning

confidence: 76%

“…Lemma 1.3. [10,14] Let G be a connected graph of order n and size m having vertex degree sequence {d 1 , d 2 , . .…”

Section: L(g)mentioning

confidence: 99%

“…Lemma 1.6. [14] Let G be a connected graph of order n and size m, where m ≥ n and let G ′ = G − e be a connected graph obtained from G by deleting an edge. Then…”

Section: L(g)mentioning

confidence: 99%

Section: L(g)mentioning

confidence: 99%

See 2 more Smart Citations

On the sum of the powers of $ A_\alpha $ eigenvalues of graphs and $ A_\alpha $-energy like invariant

Pirzada

Rather

Shaban

et al. 2022

bspm

Self Cite

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“…Theorem 2.7. [11] Let G be a connected graph with n ≥ 3 vertices, m edges, maximum degree ∆ and Zagreb index…”

Section: Introductionmentioning

confidence: 99%

Energy and Laplacian energy of unitary addition Cayley graphs

Palanivel

Chithra

2019

Filomat

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In this paper, we obtain the eigenvalues and Laplacian eigenvalues of the unitary addition Cayley graph G n and its complement. Moreover, we compute the bounds for energy and Laplacian energy for G n and its complement. In addition, we prove that G n is hyperenergetic if and only if n is odd other than the prime number and power of 3 or n is even and has at least three distinct prime factors. It is also shown that the complement of G n is hyperenergetic if and only if n has at least two distinct prime factors and n 2p.

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On Aα-spectrum of joined union of graphs and its applications to power graphs of finite groups

Rather¹,

Ganie²,

Pirzada³

2022

J. Algebra Appl.

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For a simple graph [Formula: see text], the generalized adjacency matrix [Formula: see text] is defined as [Formula: see text], where [Formula: see text] is the adjacency matrix and [Formula: see text] is the diagonal matrix of vertex degrees of [Formula: see text]. This matrix generalizes the spectral theories of the adjacency matrix and the signless Laplacian matrix of [Formula: see text]. In this paper, we find the [Formula: see text]-spectrum of the joined union of graphs in terms of the spectrum of the adjacency matrices of its components and the zeros of the characteristic polynomials of an auxiliary matrix determined by the joined union. We determine the [Formula: see text]-spectrum of join of two regular graphs, the join of a regular graph with the union of two regular graphs of distinct degrees. As applications, we investigate the [Formula: see text]-spectrum of certain power graphs of finite groups.

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On α-adjacency energy of graphs and Zagreb index

Cited by 9 publications

References 21 publications

On the sum of the powers of $ A_\alpha $ eigenvalues of graphs and $ A_\alpha $-energy like invariant

On the sum of the powers of $ A_\alpha $ eigenvalues of graphs and $ A_\alpha $-energy like invariant

Energy and Laplacian energy of unitary addition Cayley graphs

On Aα-spectrum of joined union of graphs and its applications to power graphs of finite groups

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