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