On the kernel of tree incidence matrices

Details On the kernel of tree incidence matrices On the kernel of tree incidence matrices

2000-03-03

arxiv.org/abs/cond-mat/0003049v1

Journal of Integer Sequences, Vol 3, (2000), Article 00.1.4

Physics

Condensed Matter

Statistical Mechanics

11 pages, amslatex. Also available at http://www.research.att.com/~njas/sequences/JIS/VOL3/BAUER/zerotree.html

Scientific paper

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.

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