Generalized quasirandom properties of expanding graph sequencesedding

TL;DR

This paper explores generalized quasirandom properties of expanding graph sequences, establishing equivalences and implications among spectral, discrepancy, degree, and codegree properties, which are applicable to deterministic graphs and large-scale graph embedding.

Contribution

It introduces a framework linking various spectral and combinatorial properties to generalized quasirandomness, extending classical results to multiclass and deterministic graph sequences.

Findings

Proves equivalences between spectral, discrepancy, degree, and codegree properties.

Shows these properties characterize generalized quasirandomness in deterministic graph sequences.

Supports applications in graph embedding and spectral clustering.

Abstract

We consider special multiclass spectral, discrepancy, degree, and codegree properties of expanding graph sequences. As we can prove equivalences and implications between them and the definition of the generalized quasirandomness of Lov\'asz--S\'os (2008), they can be regarded as generalized quasirandom properties akin to the equivalent quasirandom properties of the seminal Chung--Graham--Wilson paper (1989) in the one-class scenario. Since these properties are valid for certain deterministic graph sequences, irrespective of stochastic models, the partial implications also justify for law-dimensional embedding of large-scale graphs and for discrepancy minimizing spectral clustering.