Given a graph $G$ with $n$ vertices, if its adjacency matrix $A$ has eigenvalues $\lambda_1 \geq \lambda_2 \geq . . . \geq \lambda_n$ then the energy is defined as: $$E(G) =\sum_{i=1}^{n} |\lambda_i|.$$
It was introduced by Gutman in 1978 [I. Gutman, The energy of a graph, Ber. Math.-Statist. Sekt. Forschungsz. Graz 103, 1-22 (1978)). It is related to the total $\pi$-electron energy in a molecule represented by a (molecular) graph, and currently a very active research field.
Among trees with $n$ vertices, the star $S_n$ and the path $P_n$ have minimum and maximum energy, respectively.
Which graphs with $n$ vertices have minimum and maximum energy in general?
It seems that the complete graph $K_n$ does not have maximum energy.