Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Existence of Solutions to Mean Field Equations on Graphs   In this paper, we prove two existence results of solutions to mean field
equations $$\Delta u+e^u=\rho\delta_0$$ and $$\Delta u=\lambda e^u(e^u-1)+4 \pi
\sum_{j=1}^{M}{\delta_{p_j}}$$
  on an arbitrary connected finite graph, where $\rho>0$ and $\lambda>0$ are
constants, $M$ is a positive integer, and $p_1,...,p_M$ are arbitrarily chosen
vertices on the graph.

1
The categories of this paper are: cs
13
The categories of this paper are: quant-ph
27
The categories of this paper are: comp-gas
3
The categories of this paper are: physics
15
The categories of this paper are: nucl-ex
26
The categories of this paper are: bayes-an
30
The categories of this paper are: q-alg
23
The categories of this paper are: solv-int
28
The categories of this paper are: alg-geom
19
The categories of this paper are: eess
16
The categories of this paper are: stat
7
The categories of this paper are: hep-th
12
The categories of this paper are: nucl-th
37
The categories of this paper are: supr-con
36
The categories of this paper are: cmp-lg
17
The categories of this paper are: q-fin