Haemers' Minimum Rank

Haemers' minimum rank was first defined by Willem Haemers in 1979. He created this graph parameter as an upper bound for the Shannon capacity of a graph, and to answer some questions asked by Lovasz in his famous paper where he determined the Shannon capacity of a 5-cycle. In this thesis, new techniques are introduced that may be helpful for calculating Haemers' minimum rank for some graphs. These techniques are used to show the Haemers minimum rank is equal to the vertex clique cover number of a graph G for all graphs of order 10 or less, and also for some graph families, including all graphs with vertex clique cover number equal to 1, 2, 3, |G| - 2, |G| - 1, or |G|. Also, in the case of the cut-vertex reduction formula for Haemers' minimum rank, we show how this can be used to find the Shannon capacity of new graphs.</p

Techniques for determining the minimum rank of a small graph

The minimum rank of a simple graph G is defined to be the smallest possible rank over all symmetric real matrices whose ijth entry (for i≠j) is nonzero whenever {i,j} is an edge in G and is zero otherwise. Minimum rank is a difficult parameter to compute. However, there are now a number of known reduction techniques and bounds that can be programmed on a computer; we have developed a program using the open-source mathematics software Sage to implement several techniques. We have also established several additional strategies for computation of minimum rank. These techniques have been used to determine the minimum ranks of all graphs of order 7.This is a manuscript of an article from Linear Algebra and its Applications 432 (2010): 2995, doi:10.1016/j.laa.2010.01.008. Posted with permission.</p

Techniques for determining the minimum rank of a small graph

The minimum rank of a simple graph G is defined to be the smallest possible rank over all symmetric real matrices whose ijth entry (for i≠j) is nonzero whenever {i,j} is an edge in G and is zero otherwise. Minimum rank is a difficult parameter to compute. However, there are now a number of known reduction techniques and bounds that can be programmed on a computer; we have developed a program using the open-source mathematics software Sage to implement several techniques. We have also established several additional strategies for computation of minimum rank. These techniques have been used to determine the minimum ranks of all graphs of order 7.This is a manuscript of an article from Linear Algebra and its Applications 432 (2010): 2995, doi:10.1016/j.laa.2010.01.008. Posted with permission.</p