On investigations of graphs preserving the Wiener index upon vertex removal

Yi Hu; Zijiang Zhu; Pu Wu; Zehui Shao; Asfand Fahad; Yi Hu; Zijiang Zhu; Pu Wu; Zehui Shao; Asfand Fahad

doi:10.3934/math.2021750

AIMS Mathematics

2021, Volume 6, Issue 12: 12976-12985. doi: 10.3934/math.2021750

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Research article

On investigations of graphs preserving the Wiener index upon vertex removal

1.
School of Information Science and Technology, South China Business College of Guangdong University of Foreign Studies, 510545, Guangzhou, China
2.
Institute for Intelligent Information Processing, South China Business College of Guangdong University of Foreign Studies, 510545, Guangzhou, China
3.
Institute of Computing Science and Technology, Guangzhou University, Guangzhou 510006, China
4.
Department of Mathematics, COMSATS University Islamabad, Vehari Campus, Vehari 61100, Pakistan

Received: 24 October 2020 Accepted: 01 September 2021 Published: 13 September 2021
MSC : 05C10, 05C90

In this paper, we present solutions of two open problems regarding the Wiener index $ W(G) $ of a graph $ G $. More precisely, we prove that for any $ r \geq 2 $, there exist infinitely many graphs $ G $ such that $ W(G) = W(G - \{v_1, \ldots, v_r\}) $, where $ v_1, \ldots, v_r $ are $ r $ distinct vertices of $ G $. We also prove that for any $ r \geq 1 $ there exist infinitely many graphs $ G $ such that $ W(G) = W(G - \{v_i\}) $, $ 1 \leq i \leq r $, where $ v_1, \ldots, v_r $ are $ r $ distinct vertices of $ G $.
- Wiener index,
- vertex removal,
- topological index
Citation: Yi Hu, Zijiang Zhu, Pu Wu, Zehui Shao, Asfand Fahad. On investigations of graphs preserving the Wiener index upon vertex removal[J]. AIMS Mathematics, 2021, 6(12): 12976-12985. doi: 10.3934/math.2021750

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Abstract

In this paper, we present solutions of two open problems regarding the Wiener index $ W(G) $ of a graph $ G $. More precisely, we prove that for any $ r \geq 2 $, there exist infinitely many graphs $ G $ such that $ W(G) = W(G - \{v_1, \ldots, v_r\}) $, where $ v_1, \ldots, v_r $ are $ r $ distinct vertices of $ G $. We also prove that for any $ r \geq 1 $ there exist infinitely many graphs $ G $ such that $ W(G) = W(G - \{v_i\}) $, $ 1 \leq i \leq r $, where $ v_1, \ldots, v_r $ are $ r $ distinct vertices of $ G $.

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