Cospectral trees indistinguishable by scattering

TL;DR

This paper proves that for certain trees, attaching any subtree at two specific vertices results in cospectral trees only if the scattering properties at those vertices are identical, revealing a fundamental spectral indistinguishability.

Contribution

It establishes a link between cospectrality of trees with attached subtrees and identical scattering properties at specific vertices.

Findings

Cospectral trees with attached subtrees have identical scattering at certain vertices.

Attaching any tree at vertices with equal degree preserves cospectrality only if scattering functions match.

The result connects spectral properties with scattering theory in graph structures.

Abstract

Let v_1 and v_2 be two distinct vertices of a tree T_0. Let \phi_N^{(i)} (i=1,2) be the characteristic functions of the Sturm-Liouville problem on T_0 rooted at v_i with Neumann conditions at the root and let \phi_D^{(i)} (i=1,2) be the characteristic functions of the Sturm-Liouville problem on T_0 with Dirichlet conditions at the root. We prove that if attaching any tree to T_0 at the vertices v_1 and v_2 leads to cospectral trees and d(v_1)=d(v_2) then \phi_N(\lambda)^{(1)}\equiv \phi_N(\lambda)^{(2)} and \phi_D(\lambda)^{(1)}\equiv \phi_D(\lambda)^{(1)} (which means that the scattering is the same at v_1 and v_2).