The existence of the solution of the wave equation on graphs

Yong Lin; Yuanyuan Xie

REVIEW 5 major objections 5 minor 1 cited by

The existence of the solution of the wave equation on graphs

T0 review · 5 major / 5 minor · reviewed 2026-08-14 · deepseek-v4-flash

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Pith's one-line read This paper proves that the initial-boundary wave equation on a bounded domain of a locally finite weighted graph has a unique solution, and it gives an explicit eigenfunction formula for that solution.

desk verdict A routine existence theorem with a repairable gap, plus two incorrect main claims; reject as is, but not a waste of a referee's time. read the letter →

arxiv 1908.02137 v2 pith:BCKETJ4Z submitted 2019-08-06 math.AP

Yong Lin , Yuanyuan Xie This is my paper · ORCID

classification math.AP MSC 35L0535R0258J45

keywords Rothe'smethodwaveequationongraphsDirichletLaplacianinitialboundaryvalueproblemweightedgraphinfinitepropagationspeedexistenceanduniqueness

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

On a bounded piece of a locally finite weighted graph, the second-order wave equation with zero Dirichlet data is well-posed: once the interior, the boundary, and the initial position and velocity are fixed, the motion is uniquely determined. The paper proves this for forcing terms $f(t,x)$ that are Hölder continuous in time and uniformly bounded in $L^2(\Omega^\circ)$, using Rothe's method—divide time into steps, solve an elliptic problem at each step, and show that the piecewise-linear interpolants converge. The same construction leads to an explicit solution in the Dirichlet eigenbasis and to a direct consequence about speed: if the forcing is negative everywhere on the interior at time zero, the wave reaches every interior vertex immediately, so the graph wave equation has infinite propagation speed.

What carries the argument

The load-bearing mechanism is Rothe's method on a finite-dimensional state space. The time axis is split into intervals of length $\ell=T/n$; at each grid time $t_i$ one minimizes a quadratic functional whose Euler-Lagrange equation is the implicit time step $(-\Delta_\Omega u_i,v)+\ell^{-2}(u_i-2u_{i-1}+u_{i-2},v)=(f(t_i,\cdot),v)$. The piecewise-linear interpolants of the sequence $\{u_i\}$ are uniformly bounded and equi-continuous in time, so a weak limit exists and satisfies the original equation in integrated form. Uniqueness comes from the energy identity $e(t)=\int_\Omega|\nabla u|^2\,d\mu+\int_{\Omega^\circ}|\partial_t u|^2\,d\mu$, whose derivative vanishes along solutions. The eigenfunction formula uses the Dirichlet basis and the strict positivity of the eigenvalues of $-\Delta_\Omega$.

What would settle it

On the smallest graph where the issue is visible, take $\Omega^\circ=\{x_1,x_2\}$ with an edge between $x_1$ and $x_2$, each also connected to a boundary vertex, set $g=h=0$ and $f(t,x)=t\cdot\mathbf{1}_{x=x_1}$, and solve the resulting $2\times 2$ linear system exactly. Compare the exact solution with the eigenfunction formula of Theorem 1.2 and with the limit of Rothe's method on successively finer time meshes; any disagreement among the three would falsify the existence or uniqueness claim, while agreement on this domain would confirm it.

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Extended reading notes

Core claim

The paper's central claim is Theorem 1.1: if $\|f(t,\cdot)-f(s,\cdot)\|_{L^2(\Omega^\circ)}\le c|t-s|^\alpha$ and $\sup_{t\in[0,T]}\|f(t,\cdot)\|_{L^2(\Omega^\circ)}^2\le \tilde c(T)$, then the initial-boundary value problem $\partial_t^2 u-\Delta_\Omega u=f$ on $(0,\infty)\times\Omega^\circ$, with $u=0$ on the boundary and $u(0,\cdot)=g$, $\partial_t u(0,\cdot)=h$, has a unique solution: $u\in L^2([0,T];W^{1,2}_0(\Omega))$ and $\partial_t u,\partial_t^2 u\in L^2([0,T];L^2(\Omega))$ for every $T>0$. Theorem 1.2 writes that solution as a finite sum over Dirichlet eigenfunctions, with the coefficients given by sine and cosine convolutions of the forcing plus the initial modes. Theorem 1.3 reads off from the same formula that if $g=h=0$ and $f(0,x)<0$ on $\Omega^\circ$, then $u(t,x)>0$ for every interior vertex and all sufficiently small $t>0$; in other words, the wave has infinite propagation speed.

Load-bearing premise

The proof's uniqueness step assumes that the domain $\Omega$ is connected in order to infer that $\nabla u\equiv 0$ forces $u$ to be constant on $\Omega$, but Theorem 1.1 states only that $\Omega$ is a bounded domain; on a disconnected domain that inference is not justified as written, so the theorem's proof depends on an unstated connectedness hypothesis.

Editorial extensions

If this is right

For every forcing term satisfying the stated $\alpha$-Hölder condition and uniform $L^2$ bound, the graph wave equation has exactly one solution, so the initial-boundary value problem is well-posed on bounded domains of locally finite weighted graphs.
The Rothe construction gives a computable time-stepping approximation that converges to the true solution without diagonalizing the Laplacian, so it works directly from graph data such as weights and the measure $\mu$.
With zero initial data and a negative initial forcing, the solution is positive at every interior vertex for arbitrarily small positive times, so the discrete wave equation does not obey finite propagation speed.
The explicit eigenfunction formula makes the graph wave equation as explicit as the classical one: sine and cosine Duhamel integrals replace the continuum fundamental solution, and the finite sum is directly evaluable because $\Omega^\circ$ is finite.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

Because $\Omega^\circ$ is finite, the equation is a finite system of ordinary differential equations; the spectral theorem should already give well-posedness, so the Rothe proof and the eigenfunction formula can be checked against each other on any small graph.
The connectedness gap in the uniqueness proof is probably repairable: on a disconnected domain one can apply the constancy argument componentwise and use the zero boundary and zero initial data to kill every component's constant, but the theorem as stated would need the extra assumption or a revised proof.
The infinite-speed argument assumes the Dirichlet eigenfunctions are nonnegative on $\Omega^\circ$; the paper cites positivity of eigenvalues but does not prove nonnegativity of eigenfunctions. A sign-changing eigenfunction on some graph would not necessarily disprove infinite speed, but it would force a different proof of Theorem 1.3.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

The paper's main business is a Rothe-method existence proof for a linear wave equation on a finite weighted graph, followed by an explicit eigenfunction formula and a propagation-speed claim. The existence theorem is essentially standard finite-dimensional ODE well-posedness, and the proof has real gaps, but it is likely repairable. The other two theorems are wrong as stated.

Credit where it is due: the Rothe discretization is a legitimate technique, the energy estimates in Steps 3–5 are recognizable and mostly standard, and the weak-convergence argument is coherent. The literature is cited fairly, including the co-author's Poincaré inequality; self-citation is not an issue here. That part of the paper shows real effort.

The soft spots are serious, however. Theorem 1.1 is stated for any bounded domain with nonempty interior, but the uniqueness proof invokes connectedness and the existence proof uses (2.11), which fails for a component with no boundary. These are genuine proof gaps, though repairable via coercivity of the full gradient-plus-L2 form. I agree with the stress-test note here.

Theorem 1.2 is not correct: the explicit formula contains an extra term, so differentiating at t = 0 gives ∂tu(0) = h − b(0) rather than h. The derivation in Section 4 indeed arrives at ck − c̃k = −i(hk − bk(0))/√λk instead of −i hk/√λk. That is not a minor typo; it changes the solution. Theorem 1.3 then builds on that formula and also assumes an orthonormal basis of delta-type eigenfunctions (1.4), which cannot exist for a graph with edges. The sign of the propagation claim is also suspect: with g = h = 0 and f(0) < 0, the solution initially decreases, not increases. These errors are load-bearing and make the paper unacceptable in its current form.

The novelty is thin in any case: existence for a finite linear ODE system is standard, the solution formula is textbook Duhamel, and infinite propagation speed for a nonlocal graph Laplacian is immediate. For a reader working on PDE on graphs, the Rothe proof may offer some pedagogical value, but the false claims outweigh it.

Recommendation: send it to a serious referee rather than desk-reject, because the Rothe argument is substantive and the errors are identifiable and fixable in principle. But the expected outcome is rejection or major revision: Theorem 1.2 and Theorem 1.3 cannot stand as written, and Theorem 1.1 needs a corrected statement and repaired proof.

Referee Report

5 major / 5 minor

Summary. The paper studies the initial-boundary value problem for the wave equation ∂t²u−ΔΩu=f on a finite weighted graph with Dirichlet boundary data, and claims three results: Theorem 1.1 establishes existence and uniqueness by Rothe's method; Theorem 1.2 gives an explicit solution formula in an eigenfunction basis; Theorem 1.3 claims infinite propagation speed when f(0,·)<0 and the initial data vanish. The existence/uniqueness argument is a plausible finite-dimensional time-discretization, but the paper contains several errors: the explicit formula in Theorem 1.2 does not satisfy the prescribed initial velocity, Theorem 1.3 relies on an eigenfunction condition that cannot hold for a graph with edges, and the proof of Theorem 1.1 uses connectedness and a Poincaré inequality that are not part of the theorem's hypotheses.

Significance. If Theorem 1.1 were correct as stated, it would provide a useful Rothe-method proof of well-posedness for a discrete wave equation; the approach is self-contained and does not fit parameters or rely circularly on the existence results it cites. This part of the paper is likely salvageable after correcting hypotheses and estimates. However, the explicit eigenfunction formula and the infinite-propagation-speed theorem are advertised as main results and are not correct; since the paper's contribution is centered on those claims, the manuscript cannot be published in its present form.

major comments (5)

[§4, Theorem 1.2] The displayed solution does not satisfy the initial velocity condition. Differentiating the formula gives ∂tu(0,x)=Σ h_kφ_k(x) − Σ b_k(0)φ_k(x), because the two convolution terms involving the integrands cancel at t=0 and the term (1/√λ_k)(h_k−b_k(0)) sin(√λ_k t) contributes h_k−b_k(0). The manuscript instead states ∂tu(0)=Σ(b_k(0)+i(c_k−c̃_k)√λ_k)φ_k and concludes c_k−c̃_k=−i(h_k−b_k(0))/√λ_k; that step imports an erroneous b_k(0) into the initial velocity. The correct last term should be (1/√λ_k)h_k sin(√λ_k t). This is a load-bearing error because Theorem 1.3 is derived from the incorrect formula.
[§4, Theorem 1.3, assumption (1.4)] Condition (1.4) cannot be satisfied by eigenfunctions of −ΔΩ on a graph with at least one edge. If φ_k(x_j)=δ_{kj}, then at a neighbor y of x_k with y≠x_k the eigenvalue equation −ΔΩφ_k(y)=λ_kφ_k(y) gives 0 = λ_k·0 = −(1/μ(y))ω_{yx_k}(1−0) < 0 (all other φ_k(z) vanish), a contradiction. Thus Theorem 1.3 is vacuous for every nontrivial graph. Independently, its proof of ∂tu(t,x)=−Σcos(√λ_k t)b_k(0)φ_k(x) relies on the erroneous b_k(0) term of Theorem 1.2 and on the unsupported a priori bound λ_k∈(0,2).
[§3, uniqueness paragraph and Theorem 2.2] The proof of Theorem 1.1 assumes facts not stated in the theorem. Uniqueness concludes u(t,x) is constant from ∇u≡0 using 'Ω is connected', but Theorem 1.1 only assumes Ω is a bounded domain with Ω°≠∅. Existence at Step 1 invokes the Poincaré-type inequality (2.11) from Theorem 2.2, which fails if Ω has a closed component with no boundary vertices: a nonzero function constant on that component and zero elsewhere has zero gradient but nonzero L² norm. These gaps are repairable by adding a connectedness hypothesis or by arguing componentwise, but as written the theorem is broader than its proof.
[§3, Step 4, estimate (3.25)] The written bound for ‖∇u^{(n)}(t,·)‖² is not an estimate of the Rothe interpolant. In the displayed computation, the argument u_{i−1}(y)+(t−t_{i−1})u_{i−1}(y) appears instead of u_{i−1}(y)+(t−t_{i−1})δu_i(y), so the bound obtained is for ∇((1+(t−t_{i−1}))u_{i−1}) rather than for ∇(u_{i−1}+(t−t_{i−1})δu_i). The estimate can be recovered using (3.24), since ‖δu_i‖_{L²}≤√C₀, but the proof as printed does not contain that argument.
[§3, Step 6 and Definition 2.6] There is a regularity gap between the theorem and the paper's own definition of solution. Definition 2.6 requires ∂²_t u(t,x) to exist pointwise for (t,x)∈(0,∞)×Ω°, while Step 5 only proves ∂²_t u∈L²([0,T];L²(Ω)) and Step 6 obtains the equation in integrated/L² form. To match Definition 2.6 the authors need to show continuous differentiability in time, for example by using the finite-dimensional ODE structure or a time-regularity argument; as written, the existence claim is only for a weak solution.

minor comments (5)

[Title and headings] The title and several headings contain typos: 'PROPER TIES', 'W A VE EQUA TION', and 'Inﬁnite propagation speed' should be corrected.
[§3, Step 5] After stating δu^{(n_k)}(t,x)−h(x)=∫..., the next displayed identity writes (δu^{(n_k)},v)−(g,v); it should be (h,v), since the initial value of δu^{(n)} is h.
[§4, Theorem 1.3 proof] The proof asserts λ_k∈(0,2) from [4, Theorem 4.3]; for the general weighted graphs admitted by (2.5), eigenvalues of the Dirichlet Laplacian can exceed 2 when edge weights are large, so this bound needs an explicit normalization assumption.
[References] Reference [12] is a preprint without a stable identifier and [17] lacks full publication data; both should be completed.
[Theorem 1.1 statement] The Hölder condition is written 'for all x∈Ω°' although the norm already integrates over x; it should be stated as a condition on t↦f(t,·) in L²(Ω°).

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: the existence proof is a self-contained Rothe-method argument; the cited Poincare/Sobolev theorem is independent prior work, and the paper's defects are correctness gaps, not circular reductions.

full rationale

I walked the derivation chain of Theorem 1.1, Theorem 1.2, and Theorem 1.3. The uniqueness proof is an energy argument: it defines e(t) = ||grad u||^2 + ||dt u||^2, shows e'(t) = 0 using Green's formula and the wave equation itself, then concludes u = 0 from e(0) = 0. No existence result or fitted quantity is used to obtain uniqueness. The existence proof is a standard Rothe method: finite-difference minimization of F_i(u), uniform a priori bounds, weak compactness, and passage to the limit in the discrete equation. The only external input is the finite-graph Sobolev/Poincare inequality quoted as Theorem 2.2 from [5], a published theorem by Grigoryan, Lin, and Yang. Although one of the present authors is a coauthor, this is not circular: the cited theorem has stated assumptions that do not include the wave equation or any of the target conclusions, it is parameter-free, and it is independently published. It is genuine external support, not an imported uniqueness theorem used to forbid alternatives. Theorem 1.2 is a direct variation-of-constants solution of a_k'' + lambda_k a_k = b_k with initial conditions; no fitted parameter is renamed as a prediction. Theorem 1.3 is a consequence of that explicit formula under the additional assumption (1.4). I found no step in which X is defined in terms of Y, no fitted input is later called a prediction, and no load-bearing claim reduces to a self-citation chain. The paper does contain genuine correctness problems: the proof uses 'since nabla u(t,x) = 0 and Omega is connected' without stating connectedness in Theorem 1.1, and the Poincare inequality (2.11) can fail on a domain with a closed component having no boundary. There are also apparent errors in Theorem 1.2 (the h_k - b_k(0) term gives dt u(0) = h - b(0), not h) and Theorem 1.3's assumption (1.4) is generally impossible on a nontrivial graph. These are proof gaps and mathematical errors, not circularity: the claimed results are not equivalent to their inputs by construction. Therefore the circularity score is 0.

Assumptions & free parameters 0 free parameters · 4 assumptions · 0 invented entities

The paper introduces no new entities. Its free parameters are none; the proofs rely on standard graph Laplacian theory and a few unstated or incorrect assumptions.

assumptions (4)

domain assumption The graph has D_μ = max m(x)/μ(x) < ∞.
Stated in (2.5); used to make the Laplacian bounded on L² and to justify gradient estimates in Sections 2 and 3.
domain assumption The domain Ω is connected.
Used in the uniqueness proof (Section 3) to conclude ∇u ≡ 0 makes u constant on Ω. Not stated in Theorem 1.1.
domain assumption The eigenvalues λ_k of -Δ_Ω are positive and lie in (0,2).
Theorem 1.2 divides by √λ_k; Theorem 1.3 cites [4, Theorem 4.3] for λ_k ∈ (0,2). Positivity requires no zero mode, which needs each component to touch the boundary. The (0,2) bound requires a normalization that is not specified.
ad hoc to paper The eigenfunctions satisfy φ_k(x_j) = δ_{kj} and are nonnegative.
Equation (1.4) in Theorem 1.3. Delta-like eigenfunctions cannot be eigenfunctions of a graph Laplacian when any edge exists between interior vertices; general eigenfunctions also change sign.

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Pith. "Pith review of The existence of the solution of the wave equation on graphs." pith.science (2026). https://pith.science/paper/BCKETJ4Z

@misc{pith2026190802137,
  author       = {Pith},
  title        = {Pith review of: The existence of the solution of the wave equation on graphs},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/BCKETJ4Z}},
  note         = {Machine review of arXiv:1908.02137}
}

abstract

Let $G=(V, E)$ be a finite weighted graph, and $\Omega\subseteq V$ be a domain such that $\Omega^\circ\neq\emptyset$. In this paper, we study the following initial boundary problem for the non-homogenous wave equation \begin{equation*} \left\{ \begin{aligned} &\partial_t^2 u(t,x)-\Delta_\Omega u(t,x)=f(t,x),\qquad&&(t,x)\in[0,\infty)\times \Omega^\circ,\\ &u(0,x)=g(x),\qquad&& x\in\Omega^\circ,\\ &\partial_tu(0,x)=h(x),\qquad&& x\in\Omega^\circ,\\ &u(t,x)=0,\qquad&&(t,x)\in[0,\infty)\times\partial \Omega, \end{aligned} \right. \end{equation*} where $\Delta_\Omega$ denotes the Dirichlet Laplacian on $\Omega^\circ$. Using Rothe's method, we prove that the above wave equation has a unique solution.

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Forward citations

Cited by 1 Pith paper

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

Nonexistence results for the semilinear wave equation on graphs
math.AP 2025-06 reject novelty 6.0 of 10

On infinite weighted graphs, if the graph Laplacian of the distance grows slowly and suitable weighted volume growth holds, every very weak solution of the semilinear wave inequality is identically zero.

Reference graph

Works this paper leans on

18 extracted references · 18 canonical work pages · cited by 1 Pith paper

[1]

F.-C., Ngai, S.-M., Teplyaev, A.: One-dimensio nal wave equations deﬁned by fractal Laplacians

Chan, J. F.-C., Ngai, S.-M., Teplyaev, A.: One-dimensio nal wave equations deﬁned by fractal Laplacians. J. Anal. Math. 127, 219–246 (2015)

work page 2015
[2]

C.: Partial diﬀerential equations

Evans, L. C.: Partial diﬀerential equations. AMS, Provi dence R.I. (1998) 22 Y. LIN AND Y. XIE

work page 1998
[3]

Paciﬁc J

Friedman, J., Tillich, J.-P.: Wave equations for graphs and the edge-based Laplacian. Paciﬁc J. Math. 216(2), 229–266 (2004)

work page 2004
[4]

Lecture notes at Univ ersity of Bielefeld (2009)

Grigoryan, A.: Analysis on graphs. Lecture notes at Univ ersity of Bielefeld (2009)

work page 2009
[5]

Grigoryan, A., Lin, Y., Yang, Y.: Yamabe type equations o n graphs. J. Diﬀer. Equ. 261(9), 4924–4943 (2016)

work page 2016
[6]

Grigoryan, A., Lin, Y., Yang, Y.: Kazdan-Warner equatio n on graph. Calc. Var. Partial Diﬀer. Equ. 55(4), (2016)

work page 2016
[7]

Grigoryan, A., Lin, Y., Yang, Y.: Existence of positive s olutions to some nonlinear equations on locally ﬁnite graphs. Sci. China Math. 60(7), 1311–1324 (2017)

work page 2017
[8]

Haeseler, S., Keller, M., Lenz, D., Wojciechowski, R.: L aplacians on inﬁnite graphs: Dirichlet and Neumann boundary conditions. J. Spectr. Theory. 2(4), 397–432 (201 2)

work page

Show all 18 references

[9]

Academic Press, Orlando (1984), Ame r

Helgason, S.: Groups and Geometric Analysis (Integral G eometry, Invariant Diﬀerential Operators, and Spherical Functions). Academic Press, Orlando (1984), Ame r. Math. Soc., Providence (2000)

1984
[10]

Kacur, J.: Application of Rothe’s method to perturbed l inear hyperbolic equations and variational inequal- ities. Czech. Math. J. 34(1), 92–106 (1984)

1984
[11]

Waves in Random Media

Kuchment, P.: Graph models for waves in thin structures . Waves in Random Media. 12(4), 1–24 (2002)

2002
[12]

fractals

Lee, Y.-T.: Iniﬁnite propagation speed for wave soluti ons on some p.c.f. fractals. preprint, https://archive.org/details/arxiv-1111.2938

arXiv
[13]

Lin, Y., Wu, Y.: The existence and nonexistence of globa l solutions for a semilinear heat equation on graphs. Calc. Var. Partial Diﬀer. Equ. 56(4), 102 (2017)

2017
[14]

In Hyperbolic equations and related topics

Melrose, R.: Propagation for the wave group of a positiv e subelliptic second-order diﬀerential operator. In Hyperbolic equations and related topics. Academic Press, B oston, MA, (1986)

1986
[15]

Ngai, S.-M., Tang, W., Xie, Y.: Wave propagation speed o n fractals. J. Fourier Anal. Appl. to appear
[16]

Rothe, E.: Two-dimensional parabolic boundary value p roblems as a limiting case of one-dimensional bound- ary value problems. Math. Ann. 102, 650–670 (1930)

1930
[17]

Schrader, R.: Finite propagation speed and causal free quantum ﬁelds on netwoeks. J. Phys. A. Math. Gen. 42, (2009)

2009
[18]

Weber, A.: Analysis of the physical Laplacian and the he at ﬂow on a locally ﬁnite graph. J. Math. Anal. Appl. 370(1), 146–158 (2012) Yau Mathematical Sciences Center, Tsinghua University, Be ijing, 100084, China. E-mail address : yonglin@tsinghua.edu.cn College of Mathematics...

2012

Pith tools

Reviewed August 14, 2026 · model on record in the stance chip above.

[1] [1]

F.-C., Ngai, S.-M., Teplyaev, A.: One-dimensio nal wave equations deﬁned by fractal Laplacians

Chan, J. F.-C., Ngai, S.-M., Teplyaev, A.: One-dimensio nal wave equations deﬁned by fractal Laplacians. J. Anal. Math. 127, 219–246 (2015)

work page 2015

[2] [2]

C.: Partial diﬀerential equations

Evans, L. C.: Partial diﬀerential equations. AMS, Provi dence R.I. (1998) 22 Y. LIN AND Y. XIE

work page 1998

[3] [3]

Paciﬁc J

Friedman, J., Tillich, J.-P.: Wave equations for graphs and the edge-based Laplacian. Paciﬁc J. Math. 216(2), 229–266 (2004)

work page 2004

[4] [4]

Lecture notes at Univ ersity of Bielefeld (2009)

Grigoryan, A.: Analysis on graphs. Lecture notes at Univ ersity of Bielefeld (2009)

work page 2009

[5] [5]

Grigoryan, A., Lin, Y., Yang, Y.: Yamabe type equations o n graphs. J. Diﬀer. Equ. 261(9), 4924–4943 (2016)

work page 2016

[6] [6]

Grigoryan, A., Lin, Y., Yang, Y.: Kazdan-Warner equatio n on graph. Calc. Var. Partial Diﬀer. Equ. 55(4), (2016)

work page 2016

[7] [7]

Grigoryan, A., Lin, Y., Yang, Y.: Existence of positive s olutions to some nonlinear equations on locally ﬁnite graphs. Sci. China Math. 60(7), 1311–1324 (2017)

work page 2017

[8] [8]

Haeseler, S., Keller, M., Lenz, D., Wojciechowski, R.: L aplacians on inﬁnite graphs: Dirichlet and Neumann boundary conditions. J. Spectr. Theory. 2(4), 397–432 (201 2)

work page