Graphs (networks) with golden spectral ratio

TL;DR

This paper introduces the concept of golden spectral graphs, characterized by spectral ratios equal to the golden ratio, and explores their properties, existence, and relevance to real-world networks and optimal network design.

Contribution

It defines new spectral measures for graphs, introduces the concept of GSGs, and provides analytic and computational results on their properties and existence.

Findings

GSGs have good expansion and synchronizability properties.

Some real-world networks exhibit spectral ratios close to the golden ratio.

Many GSGs are Ramanujan graphs, indicating optimal expansion.

Abstract

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