Results on minimum skew rank of matrices described by a graph

Abstract

The minimum skew rank of a finite, simple, undirected graph G over a field F of characteristic not equal to 2 is defined to be the minimum possible rank of all skew-symmetric matrices over F whose i, j-entry is nonzero if and only if there exists an edge {i, j} in the graph G. The problem of determining the minimum skew rank of a graph arose after extensive study of the minimum (symmetric) rank problem. This thesis gives a background of techniques used to find minimum skew rank first developed by the IMA-ISU research group on minimum rank [9], proves cut-vertex reduction of a graph realized by a skew-symmetric matrix, and proves there is a bound for minimum skew rank created by the skew zero forcing number. The result of cut-vertex reduction is used to calculate the minimum skew ranks of families of coronas, and the minimum skew ranks of multiple other families of graphs are also computed.