BLOW-UP PROBLEMS FOR NONLINEAR PARABOLIC EQUATIONS ON LOCALLY FINITE GRAPHS

Let G =（V, E） be a locally finite connected weighted graph, and ？ be the usual graph Laplacian. In this article, we study blow-up problems for the nonlinear parabolic equation ut = ？u ＋ f（u） on G. The blow-up phenomenons for ut = ？u ＋ f（u） are discussed in terms of two cases：（i） an initial condition is given;（ii） a Dirichlet boundary condition is given. We prove that if f satisfies appropriate conditions, then the corresponding solutions will blow up in a finite time.

Mean Field Equations for the Equilibrium Turbulence and Toda Systems on Connected Finite Graphs

In this paper,we study existence of solutions of mean field equations for the equilibrium turbulence and Toda systems on connected finite graphs.Our method is based on calculus of variations,which was built on connected finite graphs by Grigor'yan,Lin and Yang.

p-Laplacian Equations on Locally Finite Graphs

This paper is mainly concerned with the following nonlinear p-Laplacian equation-△pu(x)+(λa(x)+1)|u|^(p-2)(x)u(x)=f(x,u(x)),in V on a locally finite graph G=(V,E)with more general nonlinear term,whereΔp is the discrete pLaplacian on graphs,p≥2.Under some suitable conditions on f and a(x),we can prove that the equation admits a positive solution by the Mountain Pass theorem and a ground state solution uλvia the method of Nehari manifold,for anyλ>1.In addition,asλ→+∞,we prove that the solution uλconverge to a solution of the following Dirichlet problem{-△pu(x)+|u|^(p-2)(x)u(x)=f(x,u(x)),inΩ,u(x)=0,onδΩwhereΩ={x∈V:a(x)=0}is the potential well and δΩ denotes the the boundary ofΩ.

On (t, r) Broadcast Domination of Directed Graphs

A dominating set of a graph G is a set of vertices that contains at least one endpoint of every edge on the graph. The domination number of G is the order of a minimum dominating set of G. The (t, r) broadcast domination is a generalization of domination in which a set of broadcasting vertices emits signals of strength t that decrease by 1 as they traverse each edge, and we require that every vertex in the graph receives a cumulative signal of at least r from its set of broadcasting neighbors. In this paper, we extend the study of (t, r) broadcast domination to directed graphs. Our main result explores the interval of values obtained by considering the directed (t, r) broadcast domination numbers of all orientations of a graph G. In particular, we prove that in the cases r = 1 and (t, r) = (2, 2), for every integer value in this interval, there exists an orientation of G which has directed (t, r) broadcast domination number equal to that value. We also investigate directed (t, r) broadcast domination on the finite grid graph, the star graph, the infinite grid graph, and the infinite triangular lattice graph. We conclude with some directions for future study.

Some results on entropy dimension for non-autonomous systems

In this paper,the preimage branch t-entropy and entropy dimension for nonautonomous systems are studied and some systems with preimage branch t-entropy zero are introduced.Moreover,formulas calculating the s-topological entropy of a sequence of equi-continuous monotone maps on the unit circle are given.Finally,examples to show that the entropy dimension of non-autonomous systems can be attained by any positive number s are constructed.

Existence of Three Solutions to a Class of Nonlinear Equations on Graphs

Let be a function on locally finite connect graph G=(V,E)andΩbe a bounded subset of V.We consider the nonlinear Dirichlet boundary condition problem{-△u=f(u),inΩ,u=0,onδΩ.Let f:R→R be a function satisfying certain assumptions.Then under the functional framework we use the three-solution theorem and the variational method to prove that the above equation has at least three solutions,of which one is trivial and the others are strictly positive.

Perpetual cutoff method and CDE′(K, N) condition on graphs

By using the perpetual cutoff method,we prove two discrete versions of gradient estimates for bounded Laplacian on locally finite graphs with exception sets under the condition of CDE′(K,N).This generalizes a main result of F.Münch who considers the case of CD(K,∞)curvature.Hence,we answer a question raised by Münch.For that purpose,we characterize some basic properties of radical form of the perpetual cutoff semigroup and give a weak commutation relation between bounded LaplacianΔand perpetual cutoff semigroup P w t in our setting.

On a Classical Theorem on the Diameter and Minimum Degree of a Graph

In this work, we obtain good upper bounds for the diameter of any graph in terms of its minimum degree and its order, improving a classical theorem due to Erdos, Pach, Pollack and Tuza. We use these bounds in order to study hyperbolic graphs （in the Gromov sense）. To compute the hyperbolicity constant is an almost intractable problem, thus it is natural to try to bound it in terms of some parameters of the graph. Let H（n, δ0） be the set of graphs G with n vertices and minimum degree 50, and J（n, Δ） be the set of graphs G with n vertices and maximum degree A. We study the four following extremal problems on graphs： a（n,δ0） = min{δ（G） ｜ G ∈H（n, δ0）}, b（n, δ0） =- max{δ（G）｜ e ∈H（n, δ0）}, α（n, Δ） = min{δ（G） [ G ∈ J（n, Δ）} and β（n,Δ） = max{δ（G） ] G∈Π（n,Δ）}. In particular, we obtain bounds for b（n, δ0） and we compute the precise value of a（n, δ0）, α（n, Δ） and w（n, Δ） for all values of n, r0 and A, respectively.

Erratum to “On a Classical Theorem on the Diameter and Minimum Degree of a Graph”

The original version of the article was published in [1]. Unfortunately, the original version of this article contains a mistake： in Theorem 6.2 appears that β（n, △） = （n-△ ＋ 5）/4 but the correct statement is β（n, △） = （n -△ ＋ 4）/4. In this erratum we correct the theorem and give the correct proof.

Global Gradient Estimate on Graph and Its Applications

Continuing our previous work （arXiv：1509.07981vl）, we derive another global gradient estimate for positive functions, particularly for positive solutions to the heat equation on finite or locally finite graphs. In general, the gradient estimate in the present paper is independent of our previous one. As applications, it can be used to get an upper bound and a lower bound of the heat kernel on locally finite graphs. These global gradient estimates can be compared with the Li-Yau inequality on graphs contributed by Bauer et al. [J. Differential Geom., 99, 359-409 （2015）]. In many topics, such as eigenvalue estimate and heat kernel estimate （not including the Liouville type theorems）, replacing the Li-Yau inequality by the global gradient estimate, we can get similar results.