Extreme graphs on the Sombor indices

Chenxu Yang; Meng Ji; Kinkar Chandra Das; Yaping Mao; Chenxu Yang; Meng Ji; Kinkar Chandra Das; Yaping Mao

doi:10.3934/math.20221050

AIMS Mathematics

2022, Volume 7, Issue 10: 19126-19146. doi: 10.3934/math.20221050

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Research article

Extreme graphs on the Sombor indices

1.
Department of Computer, Qinghai Normal University, Xining, Qinghai 810008, China
2.
College of Mathematical Science, Tianjin Normal University, Tianjin, China
3.
Department of Mathematics, Sungkyunkwan University, Suwon 16419, Republic of Korea
4.
Department of Mathematics, Qinghai Normal University, Xining, Qinghai 810008, China
5.
Academy of Plateau Science and Sustainability, Xining, Qinghai 810008, China

Received: 24 May 2022 Revised: 05 July 2022 Accepted: 10 July 2022 Published: 29 August 2022
MSC : 05C05, 05C12, 05C35

Gutman proposed the concept of Sombor index. It is defined via the term $ \sqrt{d_F(v_i)^2+d_F(v_j)^2} $, where $ d_F(v_i) $ is the degree of the vertex $ v_i $ in graph $ F $. Also, the reduced Sombor index and the Average Sombor index have been introduced recently, and these topological indices have good predictive potential in mathematical chemistry. In this paper, we determine the extreme molecular graphs with the maximum value of Sombor index and the extremal connected graphs with the maximum (reduced) Sombor index. Some inequalities relations among the chemistry indices are presented, these topology indices including the first Banhatti-Sombor index, the first Gourava index, the Second Gourava index, the Sum Connectivity Gourava index, Product Connectivity Gourava index, and Eccentric Connectivity index. In addition, we characterize the graph where equality occurs.
- Sombor index,
- reduced Sombor index,
- molecular graph
Citation: Chenxu Yang, Meng Ji, Kinkar Chandra Das, Yaping Mao. Extreme graphs on the Sombor indices[J]. AIMS Mathematics, 2022, 7(10): 19126-19146. doi: 10.3934/math.20221050

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Abstract

Gutman proposed the concept of Sombor index. It is defined via the term $ \sqrt{d_F(v_i)^2+d_F(v_j)^2} $, where $ d_F(v_i) $ is the degree of the vertex $ v_i $ in graph $ F $. Also, the reduced Sombor index and the Average Sombor index have been introduced recently, and these topological indices have good predictive potential in mathematical chemistry. In this paper, we determine the extreme molecular graphs with the maximum value of Sombor index and the extremal connected graphs with the maximum (reduced) Sombor index. Some inequalities relations among the chemistry indices are presented, these topology indices including the first Banhatti-Sombor index, the first Gourava index, the Second Gourava index, the Sum Connectivity Gourava index, Product Connectivity Gourava index, and Eccentric Connectivity index. In addition, we characterize the graph where equality occurs.

References

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