Quasi-randomness and Algorithmic Regularity for Graphs with General Degree Distributions

Abstract. We deal with two intimately related subjects: quasi-randomness and regular partitions. The purpose of the concept of quasi-randomness is to measure how much a given graph “resembles” a random one. Moreover, a regular partition approximates a given graph by a bounded number of quasi-random graphs. Regarding quasi-randomness, we present a new spectral characterization of low discrepancy, which extends to sparse graphs. Concerning regular partitions, we present a novel concept of regularity that takes into account the graph’s degree distribution, and show that if G = (V, E) satisfies a certain boundedness condition, then G admits a regular partition. In addition, building on the work of Alon and Naor [4], we provide an algorithm that computes a regular partition of a given (possibly sparse) graph G in polynomial time. Key words: quasi-random graphs, Laplacian eigenvalues, regularity lemma, Grothendieck’s inequality.

• Type: Conference
• Year: 2007
• Where: ICALP
• Authors: Noga Alon, Amin Coja-Oghlan, Hiêp Hàn, Mihyun Kang, Vojtech Rödl, Mathias Schacht
• Keywords: Graph’s Degree Distribution, ICALP 2007, Quasi-random Graphs, Regular Partition, Theoretical Computer Science