Connectivity of pseudomanifold graphs from an algebraic point of view

TL;DR

This paper explores the connectivity of pseudomanifold graphs through a unified algebraic framework, linking graph connectivity to graded Betti numbers of Stanley--Reisner rings, thus generalizing classical results.

Contribution

It introduces a unifying algebraic approach to analyze graph connectivity of pseudomanifolds, extending classical results and connecting to commutative algebra.

Findings

Generalized connectivity results for pseudomanifold graphs

Established a relation between graph connectivity and graded Betti numbers

Unified classical and algebraic methods in topological combinatorics

Abstract

The connectivity of graphs of simplicial and polytopal complexes is a classical subject going back at least to Steinitz, and the topic has since been studied by many authors, including Balinski, Barnette, Athanasiadis and Bjorner. In this note, we provide a unifying approach which allows us to obtain more general results. Moreover, we provide a relation to commutative algebra by relating connectivity problems to graded Betti numbers of the associated Stanley--Reisner rings.