We consider a network with cubic couplings. This is related to the well known Fermi-Pasta-Ulam-Tsingou model. We show that nonlinear periodic orbits extend from particular eigenvectors of the graph Laplacian, these are termed nonlinear normal modes. We present large classes of graphs where this occurs. These are the graphs whose Laplacian eigenvectors have components in $\{1,-1\}$ (bivalent), and $\{1,-1,0\}$ with a condition (soft-regular trivalent), the bipartite complete graphs and their extensions obtained by adding an edge between vertices having the same component. Finally, we study the stability of these solutions for chains and cycles.