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Comparison between the non-self-centrality number and the total irregularity of graphs

Author

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  • Tang, Zikai
  • Liu, Hechao
  • Luo, Huimin
  • Deng, Hanyuan

Abstract

The non-self-centrality number and the total irregularity of a connected graph G are defined as N(G)=∑|εG(vi)−εG(vj)| and irrt(G)=∑|degG(vj)−degG(vi)|, with summations embracing all pairs of vertices, degG(vi) and ɛG(vi) denoting the degree and eccentricity of the vertex vi. In this paper, we show that there exists a graph G with diameter d such that irrt(G) > N(G) for any integer d ≥ 2. This gives a complete proof of Theorem 10 in Xu et al. (2018), where Xu et al. did not prove it really for d ≥ 4. Moreover, we prove that N(T) > irrt(T) for any tree T of order n ≥ 10 with diameter d≥2+2611n and maximum degree 4 avoiding degree 3, determine all trees(unicyclic graphs) and with diameter 3 and irrt(T) > N(T) and give a sufficient condition for trees with diameter 4 and irrt(T) > N(T). These partially solve Problems 26 and 27 in the above-mentioned literature.

Suggested Citation

  • Tang, Zikai & Liu, Hechao & Luo, Huimin & Deng, Hanyuan, 2019. "Comparison between the non-self-centrality number and the total irregularity of graphs," Applied Mathematics and Computation, Elsevier, vol. 361(C), pages 332-337.
    • Handle: RePEc:eee:apmaco:v:361:y:2019:i:c:p:332-337
    DOI: 10.1016/j.amc.2019.05.054
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    References listed on IDEAS

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    1. Xu, Kexiang & Gu, Xiaoqian & Gutman, Ivan, 2018. "Relations between total irregularity and non-self-centrality of graphs," Applied Mathematics and Computation, Elsevier, vol. 337(C), pages 461-468.
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