On the structure of bidegreed graphs with minimal spectral radius

A graph is said to be (Δ, δ)-bidegreed if vertices all have one of two possible degrees: the maximum degree Δ or the minimum degree δ, with Δ, ≠ δ. We show that in the set of connected (Δ, 1)- bidegreed graphs, other than trees, with prescribed degree sequence, the graphs minimizing the adjacency matrix spectral radius cannot have vertices adjacent to Δ - 1 vertices of degree 1, that is, there are not any hanging trees. Further we consider the limit point for the spectral radius of (Δ, 1)-bidegreed graphs when degree Δ vertices are inserted in each edge between any two degree Δ vertices.