On investigations of graphs preserving the Wiener index upon vertex removal

• Main Authors: Yi Hu, Zijiang Zhu, Pu Wu, Zehui Shao, Asfand Fahad
• Format: Article
• Language: English
• Published: AIMS Press 2021-09-01
• Series: AIMS Mathematics
• Subjects: wiener index; vertex removal; topological index
• Online Access: https://www.aimspress.com/article/doi/10.3934/math.2021750?viewType=HTML
• ISSN: 2473-6988

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 $.