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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.Ìý 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.Ìý If G = (V,ÌýE) is a graph with V = {1, 2, …,Ìýn}, then a²ÔÌýnx²ÔÌýmatrixÌý´¡Ìýis a skew adjacency matrix of GÌý±è°ù´Ç±¹¾±»å±ð»åÌýa_jk = 0 whenever {j,Ìýk} is not i²ÔÌýE, and |a_jk| = 1, with a_jka_{°ìÂáÌýÌý}= -1, whenever {j,k} is i²ÔÌýE.Ìý