  We conjecture that all connected graphs of order $n$ have von Neumann entropy
at least as great as the star $K_{1,n-1}$ and prove this for almost all graphs
of order $n$. We show that connected graphs of order $n$ have R\'enyi 2-entropy
at least as great as $K_{1,n-1}$ and for $\alpha>1$, $K_n$ maximizes R\'enyi
$\alpha$-entropy over graphs of order $n$. We show that adding an edge to a
graph can lower its von Neumann entropy.
