A Discrete Variation of Littlewood--Offord Problem

TL;DR

This paper introduces a discrete variation of the Littlewood--Offord problem focusing on counting subsums that are (0,1)-vectors, and applies it to determine the maximum order of graphs with specified rank or corank.

Contribution

It presents a novel discrete version of the Littlewood--Offord problem and applies it to graph theory to find bounds on graph order based on rank and corank.

Findings

Derived bounds for the number of (0,1)-vector subsums.

Established maximum graph order given rank or corank.

Connected subsum estimates to graph adjacency matrix properties.

Abstract

Littlewood--Offord Problem concerns the number of subsums of a set of vectors that fall in a given convex set. We present a discrete variation of this problem where we estimate the number of subsums that are $(0,1)$-vectors. We then utilize this to find the maximum order of graphs with given rank or corank. The rank of a graph $G$ is the rank of its adjacency matrix $A(G)$ and the corank of $G$ is the rank of $A(G)+I$.