Extreme eigenfunctions of adjacency matrices for planar graphs employed in spatial analyses

Abstract

Mathematical properties of extreme eigenfunctions of popular geographic weights matrices used in spatial statistics are explored, and applications of these properties are presented. Three theorems are proposed and proved. These theorems pertain to the popular binary geographic weights matrix––an adjacency matrix––based upon a planar graph. They uncover relationships between the determinant of this matrix and its extreme eigenvalues, regression and the minimum eigenvalue of this matrix, and the eigenvectors of a row-standardized asymmetric version of this matrix and its symmetric similarity matrix counterpart. In addition, a conjecture is posited pertaining to estimation of the largest eigenvalue of the binary geographic weights matrix when the estimate obtained with the oldest and well-known method of matrix powering begins to oscillate between two trajectories in its convergence. An algorithm is outlined for calculating the extreme eigenvalues of geographic weights matrices based upon planar graphs. And, applications results for selected very large adjacency matrices are reported.