On the Structure of Bidegreed Graphs With Minimal Spectral Radius

Francesco Belardo

A graph is said to be $9\Delta,\delta)$-bidegreed if vertices all have one of two possible degrees: the maximum degree $\Delta$ or the minimum degree $\delta$, with $\Delta\neq\delta$. We show that in the set of connected $(\Delta,1)$-bidegreed graphs, other than trees, with prescribed degree sequence, the graphs minimizing the adjacency matrix spectral radius cannot have vertices adjacent to $\Delta-1$ vertices of degree 1, that is, there are not any hanging trees. Further we consider the limit point for the spectral radius of $(\Delta,1)$-bidegreed graphs when degree $\Delta$ vertices are inserted in each edge between any two degree $\Delta$ vertices.