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Open quipus with the same Wiener index as their quadratic line graph

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  • Ghebleh, M.
  • Kanso, A.
  • Stevanović, D.

Abstract

An open quipu is a tree constructed by attaching a pendant path to every internal vertex of a path. We show that the graph equation W(L2(T))=W(T) has infinitely many non-homeomorphic solutions among open quipus. Here W(G) and L(G) denote the Wiener index and the line graph of G respectively. This gives a positive answer to the 2004 problem of Dobrynin and Mel’nikov on the existence of solutions with arbitrarily large number of arbitrarily long pendant paths, and disproves the 2014 conjecture of Knor and Škrekovski.

Suggested Citation

  • Ghebleh, M. & Kanso, A. & Stevanović, D., 2016. "Open quipus with the same Wiener index as their quadratic line graph," Applied Mathematics and Computation, Elsevier, vol. 281(C), pages 130-136.
    • Handle: RePEc:eee:apmaco:v:281:y:2016:i:c:p:130-136
    DOI: 10.1016/j.amc.2016.01.040
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    References listed on IDEAS

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    1. Cioaba, S.M. & van Dam, E.R. & Koolen, J.H. & Lee, J.H., 2008. "Asymptotic Results on the Spectral Radius and the Diameter of Graphs," Discussion Paper 2008-71, Tilburg University, Center for Economic Research.
    2. van Dam, E.R. & Kooij, R.E., 2006. "The Minimal Spectral Radius of Graphs with a Given Diameter," Discussion Paper 2006-102, Tilburg University, Center for Economic Research.
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    Cited by:

    1. Alizadeh, Yaser & Klavžar, Sandi, 2018. "On graphs whose Wiener complexity equals their order and on Wiener index of asymmetric graphs," Applied Mathematics and Computation, Elsevier, vol. 328(C), pages 113-118.

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