The minimum rank problem for circulants

TL;DR

This paper investigates the minimum rank of circulant matrices associated with circulant graphs, establishing exact values for specific classes and exploring related parameters over real and positive semidefinite matrices.

Contribution

It introduces the minimum circulant rank, determines it for certain circulant graphs, and relates it to orthogonal representations and polynomial families, especially for prime-sized graphs.

Findings

Minimum circulant rank equals minimum rank for certain graphs.

Exact value determination for graphs with prime number of vertices.

Connection between positive semidefinite rank and orthogonal representations.

Abstract

The minimum rank problem is to determine for a graph $G$ the smallest rank of a Hermitian (or real symmetric) matrix whose off-diagonal zero-nonzero pattern is that of the adjacency matrix of $G$. Here $G$ is taken to be a circulant graph, and only circulant matrices are considered. The resulting graph parameter is termed the minimum circulant rank of the graph. This value is determined for every circulant graph in which a vertex neighborhood forms a consecutive set, and in this case is shown to coincide with the usual minimum rank. Under the additional restriction to positive semidefinite matrices, the resulting parameter is shown to be equal to the smallest number of dimensions in which the graph has an orthogonal representation with a certain symmetry property, and also to the smallest number of terms appearing among a certain family of polynomials determined by the graph. This value is then determined when the number of vertices is prime. The analogous parameter over the reals is also investigated.