Calculating and Preserving Star Sets and Star Complements of General Matrices

Abstract

This thesis presents several results relating to star sets and star complements of graphs. While a method for calculating star sets and star complements involving pro- jection matrices has been known since their introduction, a second method involving determinants is demonstrated and shown to be equivalent to the rst method. Some of the theory for star sets and star complements is expanded to general diagonalizable matrices, regardless of symmetry. The concept of preserving star sets between two general matrices is introduced and shown to be an equivalence relation, and attempts are made to classify what types of matrices can preserve the star sets of a general matrix. Finally, we determine some results for general matrices that occur when star set preservation overlaps with other equivalence relations.