On the status sequences of trees
Abstract
The status of a vertex in a connected graph is the sum of the distances from 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.
Keywords
Cite
@article{arxiv.1812.03765,
title = {On the status sequences of trees},
author = {Aida Abiad and Boris Brimkov and Alexander Grigoriev},
journal= {arXiv preprint arXiv:1812.03765},
year = {2020}
}