Sombor indices of cacti

Fan Wu; Xinhui An; Baoyindureng Wu; Fan Wu; Xinhui An; Baoyindureng Wu

doi:10.3934/math.2023078

AIMS Mathematics

2023, Volume 8, Issue 1: 1550-1565. doi: 10.3934/math.2023078

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Sombor indices of cacti

Department of Mathematics, Xinjiang University, Urumqi 830046, China

Received: 21 September 2022 Revised: 12 October 2022 Accepted: 16 October 2022 Published: 21 October 2022
MSC : 33C20, 33B15, 11B83

For a graph $ G $, the Sombor index $ SO(G) $ of $ G $ is defined as

$ SO(G) = \sum\limits_{uv\in E(G)}\sqrt{d_{G}(u)^{2}+d_{G}(v)^{2}}, $

where $ d_{G}(u) $ is the degree of the vertex $ u $ in $ G $. A cactus is a connected graph in which each block is either an edge or a cycle. Let $ \mathcal{G}(n, k) $ be the set of cacti of order $ n $ and with $ k $ cycles. Obviously, $ \mathcal{G}(n, 0) $ is the set of all trees and $ \mathcal{G}(n, 1) $ is the set of all unicyclic graphs, then the cacti of order $ n $ and with $ k(k\geq 2) $ cycles is a generalization of cycle number $ k $. In this paper, we establish a sharp upper bound for the Sombor index of a cactus in $ \mathcal{G}(n, k) $ and characterize the corresponding extremal graphs. In addition, for the case when $ n\geq 6k-3 $, we give a sharp lower bound for the Sombor index of a cactus in $ \mathcal{G}(n, k) $ and characterize the corresponding extremal graphs as well. We also propose a conjecture about the minimum value of sombor index among $ \mathcal{G}(n, k) $ when $ n \geq 3k $.
- Sombor index,
- cactus,
- extreme value
Citation: Fan Wu, Xinhui An, Baoyindureng Wu. Sombor indices of cacti[J]. AIMS Mathematics, 2023, 8(1): 1550-1565. doi: 10.3934/math.2023078

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Abstract

For a graph $ G $, the Sombor index $ SO(G) $ of $ G $ is defined as

$ SO(G) = \sum\limits_{uv\in E(G)}\sqrt{d_{G}(u)^{2}+d_{G}(v)^{2}}, $

where $ d_{G}(u) $ is the degree of the vertex $ u $ in $ G $. A cactus is a connected graph in which each block is either an edge or a cycle. Let $ \mathcal{G}(n, k) $ be the set of cacti of order $ n $ and with $ k $ cycles. Obviously, $ \mathcal{G}(n, 0) $ is the set of all trees and $ \mathcal{G}(n, 1) $ is the set of all unicyclic graphs, then the cacti of order $ n $ and with $ k(k\geq 2) $ cycles is a generalization of cycle number $ k $. In this paper, we establish a sharp upper bound for the Sombor index of a cactus in $ \mathcal{G}(n, k) $ and characterize the corresponding extremal graphs. In addition, for the case when $ n\geq 6k-3 $, we give a sharp lower bound for the Sombor index of a cactus in $ \mathcal{G}(n, k) $ and characterize the corresponding extremal graphs as well. We also propose a conjecture about the minimum value of sombor index among $ \mathcal{G}(n, k) $ when $ n \geq 3k $.

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