Graph theory, Spectra, Adjacency matrices
B. Shader and W. So introduced the idea of the skew adjacency matrix. Their idea was to give an orientation to a simple undirected graph G from which a skew adjacency matrix S(G) is created. The -adjacency matrix extends this idea to an arbitrary field F. To study the underlying undirected graph, the average -characteristic polynomial can be created by averaging the characteristic polynomials over all the possible orientations. In particular, a Harary-Sachs theorem for the average-characteristic polynomial is derived and used to determine a few features of the graph from the average-characteristic polynomial.
Hudelson, Matt; McDonald, Judi; and Wendler, Enzo.
"Alpha Adjacency: A generalization of adjacency matrices",
Electronic Journal of Linear Algebra,
Volume 35, pp. 365-375.