Generalized bilinear forms graphs and MDR codes

We investigate the generalized bilinear forms graph ${\mathrm{\Gamma }}_d$ over a residue class ring ${\mathbb{Z}}_{p^s}$. We show that ${\mathrm{\Gamma }}_d$ is a connected vertex transitive graph, and completely determine its independence number, clique number, chromatic number and maximum cliques. We also prove that cores of both ${\mathrm{\Gamma }}_d$ and its complement are maximum cliques. The graph ${\mathrm{\Gamma }}_d$ is useful for error-correcting codes. We show that every largest independent set of ${\mathrm{\Gamma }}_d$ is both an MRD code over ${\mathbb{Z}}_{p^s}$ and a usual MDS code. Moreover, there is a largest independent set of ${\mathrm{\Gamma }}_d$ to be a linear code over ${\mathbb{Z}}_{p^s}$.