On the spectra of certain rooted trees (Q819770)

The author determines the spectra of the adjacency and Laplacian matrices of simple trees \(T\) having the property that, with \(v\) being the root of \(T\), it is the union of two trees \(T_1\) and \(T_2\) for which \(V(T_1)\cap V(T_2)=\{v\}\) and \(T_1\) and \(T_2\) both have the property that the vertices in each of their levels have equal degrees. As an application, the adjacency and Laplacian spectra of starlike trees, in which lengths of any two rays differ by at most 1, are determined.