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On the Revised Szeged Index of Unicyclic Graphs with Given Diameter

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Abstract

The revised Szeged index of a connected graph G is defined as

$$\begin{aligned} Sz^*(G)=\sum _{e=uv\in E(G)}\left( n_u(e|G)+{\frac{n_0(e|G)}{2}}\right) \left( n_v(e|G)+{\frac{n_0(e|G)}{2}}\right) , \end{aligned}$$

where E(G) is the edge set of G, and for any \(e = uv\in E(G)\), \(n_u(e|G)\) is the number of vertices of G lying closer to vertex u than to v, \(n_v(e|G)\) is the number of vertices of G lying closer to vertex v than to u, and \(n_0(e|G)\) is the number of vertices with equal distances from both end vertices of the edge e. In this paper, we characterize the graph with the minimum revised Szeged index among all the unicyclic graphs with given order and diameter.

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Authors and Affiliations

  1. Department of Mathematics, Beijing Jiaotong University, Beijing, 100044, China

    Aimei Yu, Kun Peng, Rong-Xia Hao, Jiahao Fu & Yingsheng Wang

Authors
  1. Aimei Yu
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  2. Kun Peng
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  3. Rong-Xia Hao
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  4. Jiahao Fu
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  5. Yingsheng Wang
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Corresponding author

Correspondence to Aimei Yu.

Additional information

Communicated by Sanming Zhou.

The research is partially supported by Fundamental Research Funds for the Central Universities of China (No. 2015JBM107), the 111 Project of China (B16002) and the National Natural Science Foundation of China (Nos. 11771039, 11731002).

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Yu, A., Peng, K., Hao, RX. et al. On the Revised Szeged Index of Unicyclic Graphs with Given Diameter. Bull. Malays. Math. Sci. Soc. 43, 651–672 (2020). https://doi.org/10.1007/s40840-018-00706-4

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  • DOI: https://doi.org/10.1007/s40840-018-00706-4

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