In the Euclidean space setting, symmetrization inequalities is a classical theory that has been quite useful in solving problems coming from various parts of analysis: spectral geometry, variational problems, mathematical physics, spectral theory, to name a few. In my talk, I will discuss a possible extensions of this theory to the setting of graphs. It is a fairly new topic and most of the results in the area are proved in the last two years. I will talk about these developments, connections of this theory with discrete isoperimetric inequalities, and its possible applications to problems concerning 'analysis on graphs'. The talk will be at the interface of discrete math and analysis, and will be based on a joint work with Stefan Steinerberger.