The Lov\'asz number of random circulant graphs

TL;DR

This paper investigates the expected behavior of the Lovász number in dense random circulant graphs, establishing bounds and scaling laws using Fourier analysis and properties of DFT matrices.

Contribution

It provides the first bounds on the expected Lovász number for random circulant graphs, showing it scales as the square root of the number of vertices up to a log log factor.

Findings

Lovász number scales as √n up to log log factors.

Bounds on the expected Lovász number are established.

Reduction of the semidefinite program to a linear program via DFT diagonalization.

Abstract

This paper addresses the behavior of the Lov\'asz number for dense random circulant graphs. The Lov\'asz number is a well-known semidefinite programming upper bound on the independence number. Circulant graphs, an example of a Cayley graph, are highly structured vertex-transitive graphs on integers modulo $n$, where the connectivity of pairs of vertices depends only on the difference between their labels. While for random circulant graphs the asymptotics of fundamental quantities such as the clique and the chromatic number are well-understood, characterizing the exact behavior of the Lov\'asz number remains open. In this work, we provide upper and lower bounds on the expected value of the Lov\'asz number and show that it scales as the square root of the number of vertices, up to a log log factor. Our proof relies on a reduction of the semidefinite program formulation of the Lov\'asz number to a linear program with random objective and constraints via diagonalization of the adjacency matrix of a circulant graph by the discrete Fourier transform (DFT). This leads to a problem about controlling the norms of vectors with sparse Fourier coefficients, which we study using results on the restricted isometry property of subsampled DFT matrices.