Graphs (networks) with golden spectral ratio

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 spread 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, u 1⁄4 1þ ffiffiffi 5 p =2. 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 u. We have shown in this work that GSGs 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. 2007 Elsevier Ltd. All rights reserved.