Partially ordering the class of invertible trees

TL;DR

This paper introduces a partial order on invertible trees based on their inverse graphs and median eigenvalues, providing new insights into their spectral properties and extremal elements.

Contribution

It offers a new proof of Godsil's theorem and establishes a partial ordering on invertible trees, characterizing maximal and minimal elements and their spectral implications.

Findings

A partial order on invertible trees is defined based on inverse graph edges.

Maximal and minimal elements of the poset are characterized.

Median eigenvalues increase when edges are added to the inverse graph.

Abstract

A tree T is invertible if and only if T has a perfect matching. Godsil considers an invertible tree T and finds that the inverse of the adjacency matrix of T has entries in {0, 1, -1} and is the signed adjacency matrix of a graph which contains T. In this paper, we give a new proof of this theorem, which gives rise to a partial ordering relation on the class of all invertible trees on 2n vertices. In particular, we show that given an invertible tree T whose inverse graph has strictly more edges, we can remove an edge from T and add another edge to obtain an invertible tree T' whose median eigenvalue is strictly greater. This extends naturally to a partial ordering. We characterize the maximal and minimal elements of this poset and explore the implications about the median eigenvalues of invertible trees.