The Matrix Formulation of Gerrymanders: The Political Interpretation of Eigenfunctions of Connectivity Matrices

In the 1990s it is going to be easier to draw voting districts. It is also going to be harder to justify them. They are going to have to be explained in terms of increasingly sophisticated theories of fair districting. How, then, does one demonstrate a link between a particular plan and the multiple criteria for fair districting? It is proposed that a reapportionment plan can be described in terms of the eigenfunctions of a connectivity matrix. By making explicit the valencies linking nodes in a network of polygons, and/or adjusting the values in the main diagonal, the mapable eigenvectors of the connectivity matrix representing the network provide a mathematical rationale for a districting plan. This creates the possibility of simultaneously taking into account joint, and sometimes conflicting, districting criteria such as contiguity, equal population, and the protection of communities of interest.