Spectrally indistinguishable pseudorandom graphs

TL;DR

This paper constructs explicit graph families that mimic the spectral distribution of random graphs while maintaining specific structural properties like high edge density and being free of certain subgraphs.

Contribution

The authors introduce explicit graph constructions with spectral properties similar to random graphs, yet with distinct combinatorial features.

Findings

Graphs have eigenvalues following Wigner's semicircle law.

Graphs are $K_{2,3}$-free with near-maximum edge density.

Spectral indistinguishability from random graphs achieved explicitly.

Abstract

We construct explicit families of graphs whose eigenvalues are asymptotically distributed according to Wigner's semicircle law; in other words, that are spectrally indistinguishable from random graphs. However, in other respects they are strikingly dissimilar from random graphs; for example, they are $K_{2,3}$-free graphs with almost the maximum possible edge density.