We consider the Potts model with $q$ colors on a sequence of weighted graphs with adjacency matrices $A_n$, allowing for both positive and negative weights. Under a mild regularity condition the mean-field prediction for the log partition function of the Potts model on a sequence of matrices $A_n$ is asymptotically correct, whenever $\text{tr}(A_n^2)=o(n)$. In particular, our results are applicable for the Ising and the Potts models on any sequence of graphs with average degree going to $+\infty$. Using this, we establish the universality of the limiting log partition function of the ferromagnetic Potts model for a sequence of asymptotically regular graphs, and that of the Ising model for bi-regular bipartite graphs in both ferromagnetic and anti-ferromagnetic domain. We also derive a large deviation principle for the empirical measure of the colors for the Potts model on asymptotically regular graphs.