On the status sequences of trees

Aida Abiad Monge, Boris Brimkov, Alexander Grigoriev

Research output: Book/ReportReport

Abstract

The status of a vertex v in a connected graph is the sum of the distances from v to all other vertices. The status sequence of a connected graph is the list of the statuses of all the vertices of the graph. In this paper we investigate the status sequences of trees. Particularly, we show that it is NP-complete to decide whether there exists a tree that has a given sequence of integers as its status sequence. We also present some results about trees whose status sequences are comprised of a few distinct numbers or many distinct numbers. In this direction, we provide a partial answer to a conjecture of Shang and Lin from 2011, showing that any status injective tree is unique among trees. Finally, we investigate how orbit partitions and equitable partitions relate to the status sequence.
Original languageEnglish
Place of PublicationCornell University Library, US
PublisherarXiv.org at Cornell University Library
Number of pages13
Volume1812.03765
Publication statusPublished - 10 Dec 2018

JEL classifications

  • c00 - Mathematical and Quantitative Methods: General

Keywords

  • Tree
  • Status sequence
  • Status injective
  • Complexity
  • Graph partition

Cite this

Abiad Monge, A., Brimkov, B., & Grigoriev, A. (2018). On the status sequences of trees. arXiv.org at Cornell University Library. https://arxiv.org/abs/1812.03765