Signed Graph Coloring, the Art of Linear Algebra, and a Theorem of Jacobi (with Steve Morse), MathFest, July 31, 2008. PDF slides
Abstract: Signed graph colorings can be used to define a functor between specific classes of labeled graphs and multilinear functions. Under this correspondence, the “simplest” graphs correspond precisely to standard constructions of linear algebra such as the determinant, the trace, and the adjugate of a matrix. The result is a beautiful yet powerful calculus in which multilinear functions can be manipulated with ease. Using this graphical notation, we provide a one-line proof of a well-known theorem of Jacobi regarding matrix minors and determinants whose classical proof spans several pages.