Largest Laplacian Eigenvalue and Degree Sequences of Trees

TL;DR

This paper characterizes the structure of trees with the largest Laplacian eigenvalue for a given degree sequence, revealing a unique, BFS-based ordering and monotonicity properties.

Contribution

It introduces a novel structural characterization of extremal trees with maximum Laplacian eigenvalue based on degree sequences and BFS ordering.

Findings

Extremal trees have a non-increasing degree sequence in BFS order.

Maximum eigenvalue is strictly monotone with respect to majorization.

The structure of extremal trees is uniquely determined up to isomorphism.

Abstract

We investigate the structure of trees that have greatest maximum eigenvalue among all trees with a given degree sequence. We show that in such an extremal tree the degree sequence is non-increasing with respect to an ordering of the vertices that is obtained by breadth-first search. This structure is uniquely determined up to isomorphism. We also show that the maximum eigenvalue in such classes of trees is strictly monotone with respect to majorization.