Bounds of modified Sombor index, spectral radius and energy

Yufei Huang; Hechao Liu; Yufei Huang; Hechao Liu

doi:10.3934/math.2021653

AIMS Mathematics

2021, Volume 6, Issue 10: 11263-11274. doi: 10.3934/math.2021653

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Research article

Bounds of modified Sombor index, spectral radius and energy

Yufei Huang ¹,
Hechao Liu ^{2
,
,}

1.
Department of Mathematics Teaching, Guangzhou Civil Aviation College, Guangzhou, 510403, China
2.
School of Mathematical Sciences, South China Normal University, Guangzhou, 510631, China

Received: 16 March 2021 Accepted: 18 July 2021 Published: 04 August 2021
MSC : 05C09, 05C92, 05C50

Let $ G $ be a simple graph with edge set $ E(G) $. The modified Sombor index is defined as $ ^{m}SO(G) = \sum\limits_{uv\in E(G)}\frac{1}{\sqrt{d_{u}^{2}~~+~~d_{v}^{2}}} $, where $ d_{u} $ (resp. $ d_{v} $) denotes the degree of vertex $ u $ (resp. $ v $). In this paper, we determine some bounds for the modified Sombor indices of graphs with given some parameters (e.g., maximum degree $ \Delta $, minimum degree $ \delta $, diameter $ d $, girth $ g $) and the Nordhaus-Gaddum-type results. We also obtain the relationship between modified Sombor index and some other indices. At last, we obtain some bounds for the modified spectral radius and energy.
- modified Sombor index,
- modified Sombor spectral radius,
- extremal value
Citation: Yufei Huang, Hechao Liu. Bounds of modified Sombor index, spectral radius and energy[J]. AIMS Mathematics, 2021, 6(10): 11263-11274. doi: 10.3934/math.2021653

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Abstract

Let $ G $ be a simple graph with edge set $ E(G) $. The modified Sombor index is defined as $ ^{m}SO(G) = \sum\limits_{uv\in E(G)}\frac{1}{\sqrt{d_{u}^{2}~~+~~d_{v}^{2}}} $, where $ d_{u} $ (resp. $ d_{v} $) denotes the degree of vertex $ u $ (resp. $ v $). In this paper, we determine some bounds for the modified Sombor indices of graphs with given some parameters (e.g., maximum degree $ \Delta $, minimum degree $ \delta $, diameter $ d $, girth $ g $) and the Nordhaus-Gaddum-type results. We also obtain the relationship between modified Sombor index and some other indices. At last, we obtain some bounds for the modified spectral radius and energy.

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