Authors: Ghebleh, Mohammad
Kanso, Ali Adel
Stevanović, Dragan 
Title: Open quipus with the same Wiener index as their quadratic line graph
Journal: Applied Mathematics and Computation
Volume: 281
First page: 130
Last page: 136
Issue Date: 30-Apr-2016
Rank: M21
ISSN: 0096-3003
DOI: 10.1016/j.amc.2016.01.040
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.
Keywords: Graph equation | Iterated line graph | Tree | Wiener index
Publisher: Elsevier
Project: Kuwait University Research, Grant no. SM02/15

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