The Graph of the Hypersimplex

TL;DR

This paper explores the properties of the graph formed by the vertices of the (k,d)-hypersimplex, extending known graph parameters from the complete graph to these more general polytopal graphs, and discusses their combinatorial and expansion properties.

Contribution

It provides explicit formulas for graph parameters of (k,d)-hypersimplex graphs, proves their vertex transitivity and Hamilton connectivity, and analyzes their edge expansion and subgraph decomposition.

Findings

Explicit formulas for vertices, edges, degree, and diameter.

Graphs are vertex transitive and Hamilton connected.

Edge expansion rate is at least d/2.

Abstract

The (k,d)-hypersimplex is a (d-1)-dimensional polytope whose vertices are the (0,1)-vectors that sum to k. When k=1, we get a simplex whose graph is the complete graph with d vertices. Here we show how many of the well known graph parameters and attributes of the complete graph extend to a more general case. In particular we obtain explicit formulas in terms of d and k for the number of vertices, vertex degree, number of edges and the diameter. We show that the graphs are vertex transitive, hamilton connected, obtain the clique number and show how the graphs can be decomposed into self-similar subgraphs. The paper concludes with a discussion of the edge expansion rate of the graph of a (k,d)-hypersimplex which we show is at least d/2, and how this graph can be used to generate a random subset of {1,2,3,...,d} with k elements.