Energy of Graphs, Matroids and Fibonacci Numbers

The energy E(G) of a graph G is the sum of the absolute values of the eigenvalues of G. In this article we consider the problem whether generalized Fibonacci constants $\varphi_n$ $(n\geq 2)$ can be the energy of graphs. We show that $\varphi_n$ cannot be the energy of graphs. Also we prove that all natural powers of $\varphi_{2n}$ cannot be the energy of a matroid.