Note on von Neumann and Rényi entropies of a graph

We conjecture that all connected graphs of order n have von Neumann entropy at least as great as the star K1,n−1 and prove this for almost all graphs of order n. We show that connected graphs of order n have Rényi 2-entropy at least as great as K1,n−1 and for α>1, Kn maximizes Rényi α-entropy over graphs of order n. We show that adding an edge to a graph can lower its von Neumann entropy.