 
Summary: QUASIRANDOMNESS AND ALGORITHMIC REGULARITY FOR
GRAPHS WITH GENERAL DEGREE DISTRIBUTIONS
NOGA ALON, AMIN COJAOGHLAN, HI^E. P H`AN§, MIHYUN KANG¶, VOJTECH
RšODL , AND MATHIAS SCHACHT
Abstract. We deal with two intimately related subjects: quasirandomness and regular par
titions. The purpose of the concept of quasirandomness 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 quasirandom graphs. Regarding quasirandomness, we present a new spectral charac
terization of low discrepancy, which extends to sparse graphs. Concerning regular partitions, we
introduce a concept of regularity that takes into account vertex weights, 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 [Proc. 36th ACM STOC (2004) 7280], we provide an algorithm
that computes a regular partition of a given (possibly sparse) graph G in polynomial time. As an
application, we present a polynomial time approximation scheme for MAX CUT on (sparse) graphs
without "dense spots".
Key words: quasirandom graphs, Laplacian eigenvalues, regularity lemma, Grothendieck's inequal
ity.
AMS subject classification: 05C85, 05C50.
1. Introduction and Results. This paper deals with quasirandomness and
regular partitions. Loosely speaking, a graph is quasirandom if the global distribution
