Cycles in graphs and in hypergraphs: towards homology theory

TL;DR

This paper introduces algebraic topology concepts, specifically homology theory, to analyze cycles in graphs and hypergraphs, providing accessible explanations and exploring generalizations to symmetric graphs, hypergraphs, and configuration spaces.

Contribution

It presents an accessible exposition of homology theory applied to cycles in graphs and hypergraphs, including new problems and generalizations to symmetric structures and configuration spaces.

Findings

Characterization of 1-cycles in graphs

Methods to find minimal generating sets of 1-cycles

Extensions to hypergraphs and configuration spaces

Abstract

In this expository paper we present some ideas of algebraic topology (more precisely, of homology theory) in a language accessible to non-specialists in the area. A $1$-cycle in a graph is a set $C$ of edges such that every vertex is contained in an even number of edges from $C$. It is easy to check that the sum (modulo $2$) of $1$-cycles is a $1$-cycle. We start from the following problems: to find $\bullet$ the number of all $1$-cycles in a given graph; $\bullet$ a small number of $1$-cycles in a given graph such that any $1$-cycle is the sum of some of them. We consider generalizations (of these problems) to graphs with symmetry, to $2$-cycles in $2$-dimensional hypergraphs, and to certain configuration spaces of graphs (namely, to the square and the deleted square).