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

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