On the eigenvalues and eigenvectors of certain finite,vertex-weighted,bipartite graphs

The established, spectral characterisation of bipartite graphs with unweighted vertices (which are here termed homogeneous graphs) is extended to those bipartite graphs (called heterogeneous) in which all of the vertices in one set are weighted h₁ , and each of those in the other set of the bigraph is weighted h₂. All the eigenvalues of a homogeneous bipartite graph occur in pairs, around zero, while some of the eigenvalues of an arbitrary, heterogeneous graph are paired around $\text{1}\text{2}$(h₁ + h₂), the remainder having the value h₂ (or h_l). The well-documented, explicit relations between the eigenvectors belonging to “paired” eigenvalues of homogeneous graphs are extended to relate the components of the eigenvectors associated with each couple of “paired” eigenvalues of the corresponding heterogeneous graph. Details are also given of the relationships between the eigenvectors of an arbitrary, homogeneous, bipartite graph and those of its heterogeneous analogue.