The bounds of the energy and Laplacian energy of chain graphs

Yinzhen Mei; Chengxiao Guo; Mengtian Liu; Yinzhen Mei; Chengxiao Guo; Mengtian Liu

doi:10.3934/math.2021284

AIMS Mathematics

2021, Volume 6, Issue 5: 4847-4859. doi: 10.3934/math.2021284

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Research article

The bounds of the energy and Laplacian energy of chain graphs

Department of Mathematics, North University of China, Taiyuan, Shanxi 030051, China

Received: 26 November 2020 Accepted: 24 February 2021 Published: 26 February 2021
MSC : 05C50, 05C09, 05C92

Let $ G $ be a simple connected graph of order $ n $ with $ m $ edges. The energy $ \varepsilon(G) $ of $ G $ is the sum of the absolute values of all eigenvalues of the adjacency matrix $ A $. The Laplacian energy is defined as $ LE(G) = \sum_{i = 1}^{n}|\mu_{i}-\frac{2m}{n}| $, where $ \mu_{1}, \mu_{2}, \dots, \mu_{n} $ are the Laplacian eigenvalues of a graph $ G $. In this article, we obtain some upper and lower bounds on the energy and Laplacian energy of chain graph. Finally, we attain the maximal Laplacian energy among all connected bicyclic chain graphs by comparing algebraic connectivity.
- chain graph,
- energy,
- Laplacian energy,
- vertex cover number,
- algebraic connectivity
Citation: Yinzhen Mei, Chengxiao Guo, Mengtian Liu. The bounds of the energy and Laplacian energy of chain graphs[J]. AIMS Mathematics, 2021, 6(5): 4847-4859. doi: 10.3934/math.2021284

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Abstract

Let $ G $ be a simple connected graph of order $ n $ with $ m $ edges. The energy $ \varepsilon(G) $ of $ G $ is the sum of the absolute values of all eigenvalues of the adjacency matrix $ A $. The Laplacian energy is defined as $ LE(G) = \sum_{i = 1}^{n}|\mu_{i}-\frac{2m}{n}| $, where $ \mu_{1}, \mu_{2}, \dots, \mu_{n} $ are the Laplacian eigenvalues of a graph $ G $. In this article, we obtain some upper and lower bounds on the energy and Laplacian energy of chain graph. Finally, we attain the maximal Laplacian energy among all connected bicyclic chain graphs by comparing algebraic connectivity.

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