On the structure of bidegreed graphs with minimal spectral radius

Abstract

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