Optimal indexing of the vertices of graphs

Abstract:

The incidence matrices of various graphs are considered. By reordering the points, the bandwidth can be changed. In the cases of rectangular grids in the plane or cubic grids in three dimensions, the exact, minimum values of the bandwidth are determined. In certain numerical analysis problems, it is of interest to index the vertices of a graph in such a way that the matrix used to represent an associated system of linear equations is as close to diagonal as possible or, equivalently, to index the vertices of a graph in a way that minimizes the width of the band of nonzero terms in the incidence matrix for that graph. The purpose of this note is to present a few methods of determining the minimum possible width in some cases that arise in the numerical solution of Laplace’s equation. Of particular interest are the first proof of Theorem 2 which solves the problem for the common square grid and Theorem 3 which answers the problem for the cubic grid.