We generalize Schwenk's result that almost all trees contain any given limb
to trees with positive integer vertex weights. The concept of characteristic
polynomial is extended to such weighted trees and we prove almost all weighted
trees have a cospectral mate. We also prove almost all trees contain $k$
cospectral vertices for any integer $k\ge2$

Wang, Xiaojing

Yeats, Karen

English

arXiv

We generalize Schwenk's result that almost all trees contain any given limb
to trees with positive integer vertex weights. The concept of characteristic
polynomial is extended to such weighted trees and we prove that the proportion
of $n$-vertex weighted trees that is weighted cospectral to another $n$-vertex
weighted tree approaches $1$ as $n$ approaches infinity. We also prove that,
for any integer $k\ge 2$, the proportion of $n$-vertex trees containing $k$
non-similar cospectral vertices approaches $1$ as $n$ approaches infinity

Limbs and Cospectral Vertices in Trees

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