The Minimum Skew Rank of Simple Graphs to Strict Powers

Abstract

Every simple graph can be represented by an adjacency matrix. The minimum skew rank of a graph is the smallest possible rank of all skew-symmetric matrices whose non-zero entries correspond to edges of the graph. Extensive work has established methods for finding the minimum skew rank of simple graphs. The strict power of a graph, G(S) is generated by walks of exactly s steps on G. We examined the minimum skew rank of strict powers of paths using known results for n-partite complete graphs, forbidden subgraphs, algorithms for edge coverings and cut vertex reduction. We will present our method of using Wolfram Mathematica to generate the strict powers of graphs and discuss the patterns discovered in the minimum skew rank of such graphs.