Complete graphs: the space of simplicial cones, and their path tree representation

TL;DR

This paper explores the geometric structure of simplicial cones related to complete graphs, demonstrating that path tree-generated cones form a basis for the space and linking this to multivariate spline theory.

Contribution

It establishes that simplicial cones from path trees form a basis for the space of all such cones and connects geometric and combinatorial methods via spline theory.

Findings

Path tree cones form a basis for the space of simplicial cones.

Representation of cones can be analyzed through partial orders and rooted trees.

Proofs utilize multivariate spline theory.

Abstract

Let $G$ be a complete graph with $n+1$ vertices. In a recent paper of the authors, it is shown that the path trees of the graph play a special role in the structure of the truncated powers and partition functions that are associated with the graph. Motivated by the above, we take here a closer look at the geometry of the simplicial cones associated with the graph, and the role played by those simplicial cones that are generated by path trees. It is shown that the latter form a basis for the linear space spanned by the former, and that the representation of a general simplicial cone by path tree cones can be deduced by examining partial orders induced by rooted trees. While the problem itself is geometrical and its solution is combinatorial, the proofs rest with multivariate spline theory.