Brouwer degree for Chern-Simons Higgs models on finite graphs

Let $G=(V, E)$ be a finite connected graph, where $V$ denotes the set of vertices and $E$ denotes the set of edges. We revisit the following Chern-Simons Higgs model, \begin{equation*} \Delta u=\lambda \mathrm{e}^u\left(\mathrm{e}^u-1\right)+f \ \text {in} \ V, \end{equation*} where $\Delta$ is the graph Laplacian, $\lambda$ is a real number and $f$ is a function defined on $V$. Firstly, when $\lambda \int_V f \mathrm{d} \mu \neq 0$, we find that the odevity of the number of vertices in the graph affects the number of solutions. Then by calculating the topological degree and using the relationship between the degree and the critical group of a related functional, we obtain the existence of multiple solutions. Also we study the existence of solutions when $\lambda \int_V f \mathrm{d} \mu=0$. These findings extend the work of Huang et al. [Comm Math Phys 377:613-621 (2020)], Hou and Sun [Calc Var 61:139 (2022)] and Li et al. [Calc Var 63:81 (2024)]. Similarly, for the generalized Chern-Simons Higgs model, we obtain the same results. Moreover, this method is also applied to the Chern-Simons Higgs system, yielding partial results for the existence of multiple solutions. To our knowledge, this is the first instance where it has been concluded that an equation on graphs can have at least three distinct solutions. We think that our results will be valuable for studying the multiplicity of solutions to analogous equations on graphs.