  We propose two new spectral measures for graphs and networks which
characterize the ratios between the width of the "bulk" part of the spectrum
and the spectral gap, as well as the ratio between spectral length and the
width of the "bulk" part of the spectrum. Using these definitions we introduce
the concept of golden spectral graphs (GSG), which are graphs for which both
spectral ratios are identical to the golden ratio. Then, we prove several
analytic results to finding the smallest GSG as well as to build families of
GSGs. We also prove some non-existence results for certain classes of graphs.
We explore by computer several classes of graphs and found some almost GSGs.
Two networks representing real-world systems were also found to have spectral
ratios very close to the golden ratio. We have shown in this work that GSG
display good expansion properties, many of them are Ramanujan graphs and also
are expected to have very good synchronizability. In closing golden spectral
graphs are optimal networks from a topological and dynamical point of view
