α-Adjacency: A generalization of adjacency matrices

Abstract

B. Shader and W. So introduced the idea of the skew adjacency matrix.  Their idea was to give an orientation   δ to a simple undirected graph G from which a skew adjacency matrix S(G^δ) is created.  The α-adjacency matrix extends this idea to an arbitrary field F. To study the underlying undirected graph, the average α-characteristic polynomial can be created by averaging the characteristic polynomials over all the possible orientations. In particular, a Harary-Sachs theorem for the average α-characteristic polynomial is derived and used to determine a few features of the graph from the average α-characteristic polynomial.