On Minimum Terminal Distance Spectral Radius of Trees with Given Degree Sequence

TL;DR

This paper investigates the minimum terminal distance spectral radius of trees with a fixed degree sequence, proposing bounds and providing evidence that BFS-trees nearly minimize this spectral radius with high accuracy.

Contribution

It introduces bounds for the spectral radius of terminal distance matrices and supports the conjecture that BFS-trees nearly minimize this spectral radius among trees with the same degree sequence.

Findings

Lower bound based on BFS-tree's average row sum

Numerical evidence shows the gap is within 3%

Analytical steps support the conjecture's validity

Abstract

For a tree with the given sequence of vertex degrees the spectral radius of its terminal distance matrix is shown to be bounded from below by the the average row sum of the terminal distance matrix of the, so called, BFS-tree (also known as a greedy tree). This lower bound is typically not tight; nevertheless, since spectral radius of the terminal distance matrix of BFS-tree is a natural upper bound, the numeric simulation shows that relative gap between the upper and the lower bound does not exceed 3% (we also make a step towards justifying this fact analytically.) Therefore, the conjecture that BFS-tree has the minimum terminal distance spectral radius among all trees with the given degree sequence is valid with accuracy at least 97%. The same technique can be applied to the distance spectral radius of trees, which is a more popular topological index.