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Lin-Lu-Yau introduced a notion of Ricci curvature for graphs and obtained a complete classification for all Ricci-flat graphs with girth at least five. In this paper, we characterize all Ricci-flat graphs of girth four with vertex-disjoint 4-cycles.

In this paper, we prove gradient estimates for the positive solutions of $$\cal{L}u=0$$ and $$\cal{L}u = {{\partial u} \over {\partial t}}$$ on conformal solitons, where $$\cal{L}\left( \cdot \right) = \Delta \left( \cdot \right) + \left\langle {X,\nabla \left( \cdot \right)} \right\rangle$$ . We also give some applications for them.

Let $${\mathbb{D}_r}: = \{ z = x + {\rm{i}}y \in \mathbb{C}:\left| z \right| < r\} ,\,r \le 1$$ . For a normalized analytic function f in the unit disk $$\mathbb{D}: = {\mathbb{D}_1}$$ , estimating the Dirichlet integral $$\Delta (r,f) = \int {\,\int_{{\mathbb{D}_r}} {{{\left| {f\prime (z)} \right|}^2}dxdy} } $$ , z = x +iy, is an important classical problem in complex analysis. Geometrically, Δ(r, f) represents the area of the image of $${\mathbb{D}_r}$$ under f counting multiplicities. In this paper, our main objective is to estimate areas of images of $${\mathbb{D}_r}$$ under non-vanishing analytic functions of the form (z/f)μ, μ > 0, in principal powers, when f ranges over certain classes of analytic and univalent functions in $$\mathbb{D}$$ .

Let λf (n) be the normalized n-th Fourier coefficient of holomorphic eigenform f for the full modular group and ℙc(x): = {p ≤ x ∣ [pc]prime}, c ∈ ℝ+. In this paper, we show that for all 0 < c < 1 the mean value of λf(n) in ℙc(x) is ≪x log−Ax assuming the Riemann Hypothesis. Unconditionally, in the sense of Lebesgue measure, it holds for almost all c ∈ (e, 1 − e).

In this paper, we consider the following two-coupled fractional Laplacian system with two or more isolated singularities $$\left\{ {\begin{array}{*{20}{c}} {{{( - \Delta )}^s}u}&{ = {\mu _1}{u^{2q + 1}} + \beta {u^{p1 - 1}}{v^{p2}},}&\; \\ {{{( - \Delta )}^s}u}&{ = {\mu _2}{v^{2q + 1}} + \beta {u^{p1}}{v^{p2 - 1}}}&{\text{in}\;{\mathbb{R}^n}\backslash \Lambda ,} \\ {u > 0,}&{v > 0,\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;}&\; \end{array}} \right.$$ where s ∈ (0, 1), n > 2s and n ≥ 2. μ1, μ2 and β are all positive constants. p1, p2 > 1 and $${p_1} + {p_2} = 2q + 2 \in \left( {{{2n - 2s} \over {n - 2s}},\left. {{{2n} \over {n - 2s}}} \right]} \right.$$ . Λ ⊂ ℝn contains finitely many isolated points. By the method of moving plane, we obtain the symmetry results for positive solutions to above system.