Algebraic Connectivity and Degree Sequences of Trees

TL;DR

This paper characterizes trees with minimal algebraic connectivity for a given degree sequence, showing they are caterpillars with specific degree arrangements along certain paths.

Contribution

It identifies the structural properties of trees minimizing algebraic connectivity within a fixed degree sequence class, revealing they are caterpillars with non-decreasing degrees along certain paths.

Findings

Trees with minimal algebraic connectivity are caterpillars.

Vertex degrees are non-decreasing on specific paths starting at the Fiedler vector's characteristic set.

The structure of such trees is explicitly characterized.

Abstract

We investigate the structure of trees that have minimal algebraic connectivity among all trees with a given degree sequence. We show that such trees are caterpillars and that the vertex degrees are non-decreasing on every path on non-pendant vertices starting at the characteristic set of the Fiedler vector.