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The ρ‐-moments of vertex‐weighted graphs

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  • Chang, Caibing
  • Ren, Haizhen
  • Deng, Zijian
  • Deng, Bo

Abstract

Let (G,ρ) be a vertex-weighted graph of G together with the vertex set V and a function ρ(V). A ρ-moment of G at a given vertex u is defined as MGρ(u)=∑v∈Vρ(v)dist(u,v), where dist(.,.) stands for the distance function. The ρ-moment of G is the sum of moments of all vertices in G. This parameter is closely related to degree distance, Wiener index, Schultz index etc. Motivated by earlier work of Dalfo´ et al. (2013), we introduce three classes of hereditary graphs by vertex(edge)-grafting operations and give the expressions for computing their ρ-moments, by which we compute the ρ-moments of uniform(non-uniform) cactus chains and derive the order relations of ρ-moments of uniform(non-uniform) cactus chains. Based on these relations, we discuss the extremal value problems of ρ-moments in biphenyl and polycyclic hydrocarbons, and extremal polyphenyl chains, extremal spiro chains etc are given, respectively. This generalizes the results of Deng (2012).

Suggested Citation

  • Chang, Caibing & Ren, Haizhen & Deng, Zijian & Deng, Bo, 2021. "The ρ‐-moments of vertex‐weighted graphs," Applied Mathematics and Computation, Elsevier, vol. 400(C).
    • Handle: RePEc:eee:apmaco:v:400:y:2021:i:c:s0096300321001181
    DOI: 10.1016/j.amc.2021.126070
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    References listed on IDEAS

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    1. E. J. Farrell, 1987. "Matchings in hexagonal cacti," International Journal of Mathematics and Mathematical Sciences, Hindawi, vol. 10, pages 1-18, January.
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