An algorithm for generating generalized splines on graphs such as complete graphs, complete bipartite graphs and hypercubes

Abstract

An edge labeled graph is a graph G whose edges are labeled with non-zero ideals of a commutative ring R. A Generalized Spline on an edge labeled graph G is a vertex labeling of G by elements of the ring R, such that the difference between any two adjacent vertex labels belongs to the ideal corresponding to the edge joining both the vertices. The set of generalized splines forms a subring of the product ring R|V|, with respect to the operations of coordinate-wise addition and multiplication. This ring is known as the generalized spline ring RG, defined on the edge labeled graph G, for the commutative ring R. We have considered particular graphs such as complete graphs, complete bipartite graphs and hypercubes, labeling the edges with the non-zero ideals of an integral domain R and have identified the generalized spline ring RG for these graphs. Also, general algorithms have been developed to find these splines for the above mentioned graphs, for any number of vertices and Python code has been written for finding these splines.