On Order and Rank of Graphs

TL;DR

This paper investigates the maximum number of vertices in reduced graphs based on their rank, proving bounds and narrowing potential counterexamples to rank 46, thus advancing understanding of graph rank properties.

Contribution

It proves that if the conjecture on vertex bounds is false, counterexamples must have rank at most 46, and establishes an upper bound of 8 times the conjectured maximum plus 14 vertices.

Findings

Counterexamples, if any, have rank at most 46.

Every reduced graph of rank r has at most 8m(r)+14 vertices.

The conjecture holds for all ranks except possibly up to 46.

Abstract

The rank of a graph is defined to be the rank of its adjacency matrix. A graph is called reduced if it has no isolated vertices and no two vertices with the same set of neighbors. Akbari, Cameron, and Khosrovshahi conjectured that the number of vertices of every reduced graph of rank r is at most $m(r)=2^{(r+2)/2}-2$ if r is even and $m(r) = 5\cdot2^{(r-3)/2}-2$ if r is odd. In this article, we prove that if the conjecture is not true, then there would be a counterexample of rank at most $46$. We also show that every reduced graph of rank r has at most $8m(r)+14$ vertices.