  In this paper, the author considers the fractional mean field equation on a
finite graph $G=(V,E)$, say \begin{equation*} (-\Delta)^s
u=\rho\left(\dfrac{he^u}{\int_V
he^ud\mu}-\dfrac{1}{|V|}\right),\quad\forall\,x\in V, \end{equation*} where
$s\in(0,\,1)$, $\rho\in(-\infty,\,0)\cup(0,\,+\infty)$ are some fixed
parameters, $h$ denotes a given real value function on $V$. Based on the sign
of the prescribed function $h$, via the variational method, topological degree
and two mean field type heat flows, the author obtains the existence of
solutions for the above problem in three cases respectively. These results
extend the relevant research of Lin-Yang (Calc. Var., 2021), Sun-Wang (Adv.
Math., 2022) and Liu-Zhang (J. Math. Anal. Appl., 2023) in the case of $s=1$.
