For natural numbers $k<n$ we study the graphs $T_{n,k}:=K_{k}\lor\overline{K_{n-k}}$. For $k=1$, $T_{n,1}$ is the star $S_{n-1}$. For $k>1$ we refer to $T_{n,k}$ as a \emph{graph of pyramids}. We prove that the graphs of pyramids are determined by their spectrum, and that a star $S_{n}$ is determined by its spectrum iff $n$ is prime. We also show that the graphs $T_{n,k}$ are completely positive iff $k\le2$.