The maximum degree of the $r$th power of a sparse random graph

Abstract

Let $G^r_{n,p}$ denote the $r$th power of the random graph $G_{n,p}$, where $p=c/n$ for a positive constant $c$. We prove that w.h.p. the maximum degree $\Delta\left(G^r_{n,p}\right)\sim \frac{\log n}{\log_{(r+1)}n}$. Here $\log_{(k)}n$ indicates the repeated application of the log-function $k$ times. So, for example, $\log_{(3)}n=\log\log\log n$.