The biharmonic index of connected graphs

Zhen Lin; Zhen Lin

doi:10.3934/math.2022337

AIMS Mathematics

2022, Volume 7, Issue 4: 6050-6065. doi: 10.3934/math.2022337

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Research article

The biharmonic index of connected 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, Xining, 810016, Qinghai, China

Received: 19 October 2021 Revised: 07 January 2022 Accepted: 12 January 2022 Published: 17 January 2022
MSC : 05C05, 05C09, 05C35, 05C50, 05C76

Let $ G $ be a simple connected graph with the vertex set $ V(G) $ and $ d_{B}(u, v) $ be the biharmonic distance between two vertices $ u $ and $ v $ in $ G $. The biharmonic index $ BH(G) $ of $ G $ is defined as

$ BH(G) = \frac{1}{2}\sum\limits_{u\in V(G)}\sum\limits_{v\in V(G)}d_{B}^2(u, v) = n\sum\limits_{i = 2}^{n}\frac{1}{\lambda_i^2(G)}, $

where $ \lambda_i(G) $ is the $ i $-th eigenvalue of the Laplacian matrix of $ G $ with $ n $ vertices. In this paper, we provide the mathematical relationships between the biharmonic index and some classic topological indices: the first Zagreb index, the forgotten topological index and the Kirchhoff index. In addition, the extremal value on the biharmonic index for all graphs with diameter two, trees and firefly graphs are given, respectively. Finally, some graph operations on the biharmonic index are presented.
- biharmonic index,
- topological index,
- extremal value,
- graph operation
Citation: Zhen Lin. The biharmonic index of connected graphs[J]. AIMS Mathematics, 2022, 7(4): 6050-6065. doi: 10.3934/math.2022337

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Abstract

Let $ G $ be a simple connected graph with the vertex set $ V(G) $ and $ d_{B}(u, v) $ be the biharmonic distance between two vertices $ u $ and $ v $ in $ G $. The biharmonic index $ BH(G) $ of $ G $ is defined as

$ BH(G) = \frac{1}{2}\sum\limits_{u\in V(G)}\sum\limits_{v\in V(G)}d_{B}^2(u, v) = n\sum\limits_{i = 2}^{n}\frac{1}{\lambda_i^2(G)}, $

where $ \lambda_i(G) $ is the $ i $-th eigenvalue of the Laplacian matrix of $ G $ with $ n $ vertices. In this paper, we provide the mathematical relationships between the biharmonic index and some classic topological indices: the first Zagreb index, the forgotten topological index and the Kirchhoff index. In addition, the extremal value on the biharmonic index for all graphs with diameter two, trees and firefly graphs are given, respectively. Finally, some graph operations on the biharmonic index are presented.

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