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

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 $\mathrm{\Lambda }$ we associate a bipartite graph $\overline{B_{\mathrm{\Lambda }}}$ and we classify all such algebras $\mathrm{\Lambda }$ for which $\overline{B_{\mathrm{\Lambda }}}$ is regular or edge-transitive. We also show that if $\overline{B_{\mathrm{\Lambda }}}$ is semi-regular, then it is a reflexive graph.