## Abstract

The Grothendieck constant of a graph G = (V, E) is the least constant K such that for every matrix A : V × V → R, under(max, f : V → S^{| V | - 1}) under(∑, {u, v} ∈ E) A (u, v) {dot operator} 〈 f (u), f (v) 〉 ≤ K under(max, ε{lunate} : V → {- 1, + 1}) under(∑, {u, v} ∈ E) A (u, v) {dot operator} ε{lunate} (u) ε{lunate} (v) . The investigation of this parameter, introduced in [N. Alon, K. Makarychev, Y. Makarychev, A. Naor, Quadratic forms on graphs, in: Proc. of the 37th ACM STOC, ACM Press, Baltimore, 2005, pp. 486-493 (Also: Invent. Math. 163 (2006) 499-522)], is motivated by the algorithmic problem of maximizing the quadratic form ∑_{{u, v} ∈ E} A (u, v) ε{lunate} (u) ε{lunate} (v) over all ε{lunate} : 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, Θ (log (n p)). This settles a problem raised in [N. Alon, K. Makarychev, Y. Makarychev, A. Naor, Quadratic forms on graphs, in: Proc. of the 37th ACM STOC, ACM Press, Baltimore, 2005, pp. 486-493 (Also: Invent. Math. 163 (2006) 499-522)]. We also obtain a similar estimate for regular graphs in which the absolute value of each nontrivial eigenvalue is small.

Original language | English |
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Pages (from-to) | 323-327 |

Number of pages | 5 |

Journal | Discrete Optimization |

Volume | 5 |

Issue number | 2 |

DOIs | |

State | Published - May 2008 |

### Bibliographical note

Funding Information:The first author’s research was supported in part by the Israel Science Foundation, by a USA–Israel BSF grant, and by the Hermann Minkowski Minerva Center for Geometry at Tel Aviv University.

## Keywords

- Grothendieck constant of a graph
- Random graphs
- Semidefinite programming

## ASJC Scopus subject areas

- Theoretical Computer Science
- Computational Theory and Mathematics
- Applied Mathematics