Sharp bounds on the zeroth-order general Randić index of trees in terms of domination number

Chang Liu; Jianping Li; Chang Liu; Jianping Li

doi:10.3934/math.2022142

AIMS Mathematics

2022, Volume 7, Issue 2: 2529-2542. doi: 10.3934/math.2022142

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Research article

Sharp bounds on the zeroth-order general Randić index of trees in terms of domination number

Chang Liu ,
Jianping Li ^,

College of Liberal Arts and Sciences, National University of Defense Technology, Changsha 410073, China

Received: 20 July 2021 Accepted: 08 November 2021 Published: 16 November 2021
MSC : 05C05, 05C35, 05C69

The zeroth-order general Randić index of graph $ G = (V_G, E_G) $, denoted by $ ^0R_{\alpha}(G) $, is the sum of items $ (d_{v})^{\alpha} $ over all vertices $ v\in V_G $, where $ \alpha $ is a pertinently chosen real number. In this paper, we obtain the sharp upper and lower bounds on $ ^0R_{\alpha} $ of trees with a given domination number $ \gamma $, for $ \alpha\in(-\infty, 0)\cup(1, \infty) $ and $ \alpha\in(0, 1) $, respectively. The corresponding extremal graphs of these bounds are also characterized.
- the zeroth-order general Randić index,
- extremal trees,
- domination number
Citation: Chang Liu, Jianping Li. Sharp bounds on the zeroth-order general Randić index of trees in terms of domination number[J]. AIMS Mathematics, 2022, 7(2): 2529-2542. doi: 10.3934/math.2022142

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Abstract

The zeroth-order general Randić index of graph $ G = (V_G, E_G) $, denoted by $ ^0R_{\alpha}(G) $, is the sum of items $ (d_{v})^{\alpha} $ over all vertices $ v\in V_G $, where $ \alpha $ is a pertinently chosen real number. In this paper, we obtain the sharp upper and lower bounds on $ ^0R_{\alpha} $ of trees with a given domination number $ \gamma $, for $ \alpha\in(-\infty, 0)\cup(1, \infty) $ and $ \alpha\in(0, 1) $, respectively. The corresponding extremal graphs of these bounds are also characterized.

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