Algebraic Graph Theory Seminar - Judi McDonald

Title: Skew Adjacency Matrices:  The Number of Characteristic polynomials of Skew Cacti

Abstract:

In this talk, I will begin by reviewing the Harry-Sachs method for finding the characteristic polynomial of a matrix with a given digraph, as well as a couple of useful variations.  Then I will illustrate how we (JMcD, Matt Hudelson, Amy Streifel) used this technique to explore the number of different characteristic polynomials that can be achieved by skew adjacency matrices of a given graph.  If G = (V, E) is a graph with V = {1, 2, …, n}, then an nxn matrix A is a skew adjacency matrix of G provided a_jk = 0 whenever {j, k} is not in E, and |a_jk| = 1, with a_jka_kj  = -1, whenever {j,k} is in E.