On the sum of the largest $ A_{\alpha} $-eigenvalues of graphs

Zhen Lin; Zhen Lin

doi:10.3934/math.2022825

AIMS Mathematics

2022, Volume 7, Issue 8: 15064-15074. doi: 10.3934/math.2022825

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Research article

On the sum of the largest $ A_{\alpha} $-eigenvalues of graphs

Zhen Lin ^{1,2
,
,}

1.
School of Mathematics and Statistics, The State Key Laboratory of Tibetan Intelligent Information Processing and Application, Qinghai Normal University, Xining 810008, Qinghai, China
2.
Academy of Plateau Science and Sustainability, People's Government of Qinghai Province and Beijing Normal University, Xining 810016, Qinghai, China

Received: 01 March 2022 Revised: 01 May 2022 Accepted: 30 May 2022 Published: 15 June 2022
MSC : 05C50, 05C09, 05C90

Let $ A(G) $ and $ D(G) $ be the adjacency matrix and the degree diagonal matrix of a graph $ G $, respectively. For any real number $ \alpha \in[0, 1] $, Nikiforov defined the $ A_{\alpha} $-matrix of a graph $ G $ as $ A_{\alpha}(G) = \alpha D(G)+(1-\alpha)A(G) $. Let $ S_k(A_{\alpha}(G)) $ be the sum of the $ k $ largest eigenvalues of $ A_{\alpha}(G) $. In this paper, some bounds on $ S_k(A_{\alpha}(G)) $ are obtained, which not only extends the results of the sum of the $ k $ largest eigenvalues of the adjacency matrix and signless Laplacian matrix, but it also gives new bounds on graph energy.
- $ A_{\alpha} $-matrix,
- sum of $ A_{\alpha} $-eigenvalues,
- energy,
- graph operation
Citation: Zhen Lin. On the sum of the largest $ A_{\alpha} $-eigenvalues of graphs[J]. AIMS Mathematics, 2022, 7(8): 15064-15074. doi: 10.3934/math.2022825

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Abstract

Let $ A(G) $ and $ D(G) $ be the adjacency matrix and the degree diagonal matrix of a graph $ G $, respectively. For any real number $ \alpha \in[0, 1] $, Nikiforov defined the $ A_{\alpha} $-matrix of a graph $ G $ as $ A_{\alpha}(G) = \alpha D(G)+(1-\alpha)A(G) $. Let $ S_k(A_{\alpha}(G)) $ be the sum of the $ k $ largest eigenvalues of $ A_{\alpha}(G) $. In this paper, some bounds on $ S_k(A_{\alpha}(G)) $ are obtained, which not only extends the results of the sum of the $ k $ largest eigenvalues of the adjacency matrix and signless Laplacian matrix, but it also gives new bounds on graph energy.

References

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