The spectrum of asymptotic Cayley trees (Q6562975)

A regular Cayley tree of degree \(q\) is an infinite tree graph \(T_q\) where all vertices have degree \(q\). A planted Cayley tree of degree \(q\) is an infinite tree graph \(T_q^\prime\) where all vertices except one, called the root vertex \(r\), have degree \(q\). An asymptotic Cayley tree of degree \(q\) is a graph \(G\) which can be obtained by grafting a finite number of planted Cayley trees of degree \(q\) onto a finite graph \(B\) (here, by grafting, it is meant that the root vertex \(r\) of \(T_q^\prime\) is identified with a vertex of \(B\)). The unique minimal such finite graph \(B\) is called the core of \(G\).\N\NIn this paper, three theorems are proved about the spectrum of asymptotic Cayley trees of degree \(q\). The first theorem is that the spectrum of such graphs consists of an absolutely continuous part and a finite set of eigenvalues (the pure point spectrum). The second theorem shows that the eigenvalues in the pure point spectrum have finite multiplicity; furthermore, bounds are given on the multiplicities. The third theorem bounds the total dimension of the eigenspaces of an asymptotic Cayley tree of degree \(q\) in terms of the total dimension of eigenspaces of the core of the same asymptotic Cayley tree.\N\NIn the remainder of the paper, some examples of the computation of the pure point spectrum of asymptotic Cayley trees are given and the results on the spectrum of asymptotic Cayley trees are used to study the properties of quantum walks on the same asymptotic Cayley trees.