In this paper, we focus on connected locally finite graphs G = (V, E). First, we assume that there are two constants μ_0 and ω_0, which make the measure function and symmetric weight function satisfy μ(x) ≥ μ_0 ∀, x ∈ V, ω_{xy} ≥ ω_0, ∀xy ∈ E. Based on this assumption, we obtain two interesting embedding theorems on finite graphs: W_0^{1,2} (B_k)↪L^p (B_k), W^{1,2} (B_k)↪L^p (B_k). Although their inclusion relations are obvious on finite graphs, here we mainly give the control relations under the same control coefficient. Secondly, Δ is the Laplace operator on a general graph. Due to Lin and Yang (2022), using calculus of variations from local to global, we establish the existence of solutions to the exponential power type nonlinear Schrödinger equation, says −Δu + hu = f u e^{u^2} + g, x ∈ V, and the existence of solutions for fractional nonlinear mean field equations, says −Δu + hu = (g e^u) / (∫_V g e^u dμ) + f /(u + m), x ∈ V. When f, g and h satisfy some conditions, we prove the existence of non explicit solutions for the above two kinds of equations in a specific space.