Burnside graphs, algebras generated by sets of matrices, and the Kippenhahn Conjecture

Given a set of matrices, it is often of interest to determine the algebra they generate. Here we exploit the concept of the Burnside graph of a set of matrices, and show how it may be used to deduce properties of the algebra they generate. We prove two conditions regarding a set of matrices generating the full algebra; the first necessary, the second sufficient. An application of these results is given in the form of a new family of counterexamples to the Kippenhahn conjecture, of order $8 \times 8$ and greater.