On the inverse eigenvalue problem for block graphs

Abstract

In this work, the inverse eigenvalue problem is completely solved for a subfamily of clique-path graphs, in particular for lollipop graphs and generalized barbell graphs. For a matrix A with associated graph G, a new technique utilizing the strong spectral property is introduced, allowing us to construct a matrix A′ whose graph is obtained from G by appending a clique while arbitrary list of eigenvalues is added to the spectrum. Consequently, many spectra are shown realizable for block graphs.