In this article, we consider the Einstein-scalar field Lichnerowicz equation

$$\begin{aligned} -\Delta u+hu=Bu^{p-1}+Au^{-p-1}\end{aligned}

on any connected finite graph G = (V, E), where A, B, h are given functions on V with A ≥ 0, A ≡ 0 on V, and p > 2 is a constant. By using the classical variational method, topological degree theory and heat-flow method, we provide a systematical study on this equation by providing the existence results for each case: positive, negative and null Yamabe-scalar field conformal invariant, namely h > 0, h < 0 and h = 0 respectively.