Spectral ordering and 2-switch transformations

TL;DR

This paper investigates how to order trees with identical degree sequences based on their spectral radii, using 2-switch transformations to establish a total order within a specific family.

Contribution

It introduces a method to determine a total ordering of trees by spectral radius using 2-switch transformations and characterizes when the spectral index decreases.

Findings

Established conditions for spectral index decrease under 2-switches.

Determined a total ordering for a family of trees based on spectral radii.

Provided insights into spectral properties related to tree size parameters.

Abstract

We address the problem of ordering trees with the same degree sequence by their spectral radii. To achieve that, we consider 2-switch transformations which preserve the degree sequence and establish when the index decreases. Our main contribution is to determine a total ordering of a particular family by their indices according to a given parameter related to sizes in the tree.