A modified Grassmann algebra approach to theorems on permanents and determinants

Abstract Abdessalem [1] showed how the techniques of Grassmann-Berezin calculus can be used to prove the all-minors matrix-tree theorem. Starting with any matrix A we associate a directed graph G to A equipped with two different weight functions on the edges. We extend the approach of Abdessalem to give interpretations for the permanents and determinants of all square sub-matrices of A in terms of weighted counts of sub-graphs of G using each of the edge-weight functions. We introduce a modification of Grassmann-Berezin variables to obtain our results.