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

Zhen Lin; Zhen Lin

doi:10.3934/mmc.2022007

Mathematical Modelling and Control

2022, Volume 2, Issue 2: 55-64. doi: 10.3934/mmc.2022007

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Research article

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

Zhen Lin ^{1,2
,
,}

1.
School of Mathematics and Statistics, Qinghai Normal University, Xining, 810008, Qinghai, China
2.
Academy of Plateau Science and Sustainability, People's Government of Qinghai Province and Beijing Normal University, China

Received: 05 November 2021 Revised: 04 March 2022 Accepted: 03 April 2022 Published: 27 June 2022

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 recently defined the $ A_{\alpha} $-matrix of $ G $ as $ A_{\alpha}(G) = \alpha D(G)+(1-\alpha)A(G) $. The graph invariant $ S_{\alpha}^{p}(G) $ is the sum of the $ p $-th power of the $ A_{\alpha} $-eigenvalues of $ G $ for $ \frac{1}{2} < \alpha < 1 $, which has a close relation to the $ \alpha $-Estrada index. In this paper, we establish some bounds on $ S_{\alpha}^{p}(G) $ and characterize the extremal graphs. In particular, we present some bounds on $ S_{\alpha}^{p}(G) $ in terms of the degree sequences, order and size of $ G $ by using majorization techniques. Moreover, we give lower and upper bounds for $ S_{\alpha}^{p}(G) $ of a bipartite graph and characterize the extremal graphs.
- $ A_{\alpha} $-matrix,
- $ A_{\alpha} $-eigenvalues,
- majorization
Citation: Zhen Lin. On the sum of powers of the $ A_{\alpha} $-eigenvalues of graphs[J]. Mathematical Modelling and Control, 2022, 2(2): 55-64. doi: 10.3934/mmc.2022007

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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 recently defined the $ A_{\alpha} $-matrix of $ G $ as $ A_{\alpha}(G) = \alpha D(G)+(1-\alpha)A(G) $. The graph invariant $ S_{\alpha}^{p}(G) $ is the sum of the $ p $-th power of the $ A_{\alpha} $-eigenvalues of $ G $ for $ \frac{1}{2} < \alpha < 1 $, which has a close relation to the $ \alpha $-Estrada index. In this paper, we establish some bounds on $ S_{\alpha}^{p}(G) $ and characterize the extremal graphs. In particular, we present some bounds on $ S_{\alpha}^{p}(G) $ in terms of the degree sequences, order and size of $ G $ by using majorization techniques. Moreover, we give lower and upper bounds for $ S_{\alpha}^{p}(G) $ of a bipartite graph and characterize the extremal graphs.

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