On the minimum rank of a graph over finite fields

Shmuel Friedland,Raphael Loewy

doi:10.1016/j.laa.2011.06.041

Abstract

In this paper we deal with two aspects of the minimum rank of a simple undirected graph G on n vertices over a finite field Fq with q elements, which is denoted by mr(Fq,G). In the first part of this paper we show that the average minimum rank of simple undirected labeled graphs on n vertices over F2 is (1-εn)n, were limn→∞εn=0.In the second part of this paper we assume that G contains a clique Kk on k-vertices. We show that if q is not a prime then mr(Fq,G)⩽n-k+1 for 4⩽k⩽n-1 and n⩾5. It is known that mr(Fq,G)⩽3 for k=n-2, n⩾4 and q⩾4. We show that for k=n-2 and each n⩾10 there exists a graph G such that mr(F3,G)>3. For k=n-3, n⩾5 and q⩾4 we show that mr(Fq,G)⩽4.

Highlights

Let G = (V, E) be a simple undirected graph on the set of vertices V and the set of edges E
The second aspect of this problem is estimating the minimum rank of graphs which contain a clique
As for F2, let n ≥ 5 and consider the graph G obtained from Kn−1 by adding a new vertex v and connecting it to exactly two of the vertices of Kn−1

Summary

Introduction

The second aspect of this problem is estimating the minimum rank of graphs which contain a clique. As for F2, let n ≥ 5 and consider the graph G obtained from Kn−1 by adding a new vertex v and connecting it to exactly two of the vertices of Kn−1. It follows from [BvdHL] that mr(F2, G) = 3.

Scaled average minimum rank over F2

Infinite fields