In this paper, we study the Mean field equation and the relativistic Abelian Chern-Simons equations (involving two Higgs particles and any two gauge fields) on the finite connected graphs. For the former equation, we establish the existence results and some uniqueness result. In particular, we find that there is no set of critical parameters for the Mean field equation on the finite graphs and the existence is ensured for any non-negative parameters, which is in contrast to the continuous case. In addition, we give the optimal constant which is the threshold for the uniqueness of the equation on the finite complete graphs with simple weight. A key observation is that the solution can take at most two values. While for the second problem, we study the existence of maximal condensates, and also establish the existence of multiple solutions, including a local minimizer for the transformed energy functional and a mountain-pass type solution.

在本文中，我们研究了有限连通图上的平均场方程和相对论阿贝尔陈-西蒙斯方程（涉及两个希格斯粒子和任意两个规范场）。对于前一个方程，我们建立了存在性结果和一些唯一性结果。特别地，我们发现有限图上的平均场方程没有一组关键参数，并且任何非负参数都确保存在，这与连续情况相反。此外，我们给出了最优常数，它是具有简单权重的有限完全图上方程唯一性的阈值。一个关键的观察结果是解决方案最多可以采用两个值。而对于第二个问题，我们研究了极大凝聚的存在性，同时也建立了多重解的存在性，