On the kernel of tree incidence matrices

TL;DR

This paper analyzes the spectral properties of random tree incidence matrices, particularly focusing on the eigenvalue zero, providing asymptotic fractions, closed forms, and generating functions for their multiplicities.

Contribution

It introduces a precise asymptotic estimate for the eigenvalue zero fraction in large random trees and derives closed-form and generating function expressions for finite trees.

Findings

Average eigenvalue zero fraction asymptotic to 0.1343

Closed form and generating function for total zero eigenvalue multiplicity

Asymptotic estimate for large random trees

Abstract

We study the height of the delta peak at 0 in the spectrum of random tree incidence matrices. We show that the average fraction of the spectrum occupied by the eigenvalue 0 in a large random tree is asymptotic to 2x-1 = 0.1342865808195677459999... where x is the unique real root of x = exp(-x). For finite trees, we give a closed form, a generating function, and an asymptotic estimate for the sequence 1,0,3,8,135,1164,21035,.... of the total multiplicity of the eigenvalue 0 in the set of n^{n-2} tree incidence matrices of size n>0.