 
Summary: The Grothendieck constant of random and pseudorandom graphs
To the memory of George Dantzig
Noga Alon
Eli Berger
Abstract
The Grothendieck constant of a graph G = (V, E) is the least constant K such that for every
matrix A : V × V R:
max
f:V SV 1
{u,v}E
A(u, v) · f(u), f(v) K max
:V {1,+1}
{u,v}E
A(u, v) · (u) (v).
The investigation of this parameter, introduced in [2], is motivated by the algorithmic problem of
maximizing the quadratic form {u,v}E A(u, v) (u) (v) over all : V {1, 1}, which arises in
the study of correlation clustering and in the investigation of the spin glass model. In the present
note we show that for the random graph G(n, p) the value of this parameter is, almost surely,
(log(np)). This settles a problem raised in [2]. We also obtain a similar estimate for regular
graphs in which the absolute value of each nontrivial eigenvalue is small.
