Higher homological algebra for one-point extensions of bipartite hereditary algebras and spectral graph theory

TL;DR

This paper explores the connection between higher homological algebra of certain algebras and spectral graph theory, classifying specific algebra types based on properties of associated bipartite graphs.

Contribution

It introduces a classification of 2-representation-finite quadratic monomial algebras via properties of their associated bipartite graphs, linking algebraic and graph-theoretic concepts.

Findings

Classification of algebras with regular bipartite graphs

Identification of edge-transitive bipartite graphs associated with algebras

Characterization of semi-regular graphs as reflexive graphs

Abstract

In this article we study higher homological properties of $n$-levelled algebras and connect them to properties of the underlying graphs. Notably, to each $2$-representation-finite quadratic monomial algebra $\Lambda$ we associate a bipartite graph $\overline{B_{\Lambda}}$ and we classify all such algebras $\Lambda$ for which $\overline{B_{\Lambda}}$ is regular or edge-transitive. We also show that if $\overline{B_{\Lambda}}$ is semi-regular, then it is a reflexive graph.