The Simplex Geometry of Graphs

TL;DR

This paper introduces a novel geometric approach using simplex geometry to analyze graphs, establishing a correspondence between graph properties and simplex characteristics, offering new insights and potential applications in graph theory.

Contribution

It presents the first formal connection between graph theory and simplex geometry, expanding analytical tools for studying graphs.

Findings

Established a graph-simplex correspondence

Linked graph characteristics to simplex properties

Suggested potential applications in graph analysis

Abstract

Graphs are a central object of study in various scientific fields, such as discrete mathematics, theoretical computer science and network science. These graphs are typically studied using combinatorial, algebraic or probabilistic methods, each of which highlights the properties of graphs in a unique way. Here, we discuss a novel approach to study graphs: the simplex geometry (a simplex is a generalized triangle). This perspective, proposed by Miroslav Fiedler, introduces techniques from (simplex) geometry into the field of graph theory and conversely, via an exact correspondence. We introduce this graph-simplex correspondence, identify a number of basic connections between graph characteristics and simplex properties, and suggest some applications as example.