Equivalent Properties for CD Inequalities on Graphs with Unbounded Laplacians

Chao GONG Yong LIN

1 Introduction

Graph theory is the basic theory of the study of graphs and networks.The spectral graph theory,which is used for describing the structure and characteristic of graphs by adjacency matrix or the spectral density of Laplacian matrix,is the classical method for studying graph(see[1]).

We have already known that we can find the curvature by some way,and there are many examples for the geometric analysis such as the famous Li-Yau gradient estimate.Moreover,we can use some data to describe the graphs and optimize it such as the Cheeger constant on graphs.

The Laplacian on graph has always been an important research topic.In fact,the Laplacian can be seen as the generator of symmetric Markov process.Laplacians always appear in the topics on the research of discrete structure for heat equations as in[2].

As for the Laplacians on graphs,the properties are different on different occasions,such as finite graphs,locally finite graphs and in finite graphs.If we assume that the graph is finite,then the properties of Laplacians are simple and good.But for some problems,the assumption of finite graph is obviously too narrow,so locally finite graphs or some in finite graphs can be a better research object.We still can get good enough properties on them.In recent years,some research topics are as follows on Laplacians on in finite graphs:

(a)Definition of the operators and essential selfadjointness.

(b)Absence of essential spectrum.

(c)Stochastic incompleteness.

The results on metric space can be seen in[3]and meanwhile,it also has a similar geometric structure as the manifold.Obviously,the graph can also be seen as a kind of metric space.We can define the distance between two vertices of the graph as the natural metric,which is the number of the minimum edges connecting them.Then,we should consider whether the theories of Riemannian manifold can be extended to the graph,especially those about Ricci curvature.Many results in geometry analysis come from the Ricci curvature,especially the lower bound of Ricci curvature,such as the heat kernel estimation,Harnack inequalities and Sobolev inequalities.These conclusions have been made in[4].

On Riemannian manifolds,There exists an identical Bochner formula for any smooth function:

When its Ricci curvature has a lower bound,we can make a conclusion that for any η∈TM there exists a K ∈ R that satisfies.Unfortunately in the discrete situation we can not defineBut on Riemannian manifolds we can make use of Cauchy-Schwarz inequalities to get an inequalityThen the Bochner inequality can be rewritten into

The inequality above is the so-called curvature-dimension inequality on Riemannian manifolds,and we call it CD inequality for short.Using this inequality,the ”Ricci curvature” in the discrete situation can be defined.Bakery and Emery have already proved that if the chain rule is satisfied,the CD inequalities can be extended to the Markov operators on some metric space.Yet obviously the chain rule is not always true for discrete functions.Fortunately whenupsatisfies the chain rule even on the discrete condition.So,[4]introduced an improved CD inequality—CDE inequality.This definitely is a key for the research of the discrete geometry analysis.

This paper gives an introduction of the CD inequality and several equivalent conditions of the CD inequality for unbounded Laplacians on the graph.It is organized into three parts.

Chapter 1 gives the introduction of the graph,the Laplacians and CD inequalities on it.

Chapter 2 introduces some basic conclusions in order to get the main result,and some definitions such as the locally finite graph,the weighted graph and the domain of the operators.

Chapter 3 gives the main conclusion of this paper which includes some equivalent conditions of the CD inequalities.

2 Graphs,Laplacians and CD Inequalities

Given a graph G=(V,E),for an x∈V,if there exists another y∈V that satisfies(x,y)∈E,we call them neighbors,and write as x～y.If there exists an x∈V satisfying(x,x)∈E,we call it a self-loop.In this paper we allow graphs to have self-loops.

Now we will introduce some basic definitions and theorems before we get the main results.

Definition 2.1(Locally Finite Graph)We call a graph G a locally finite graph if for any x∈V,it satisfies#{y∈V|y～x}<∞.Moreover,it is called connected if for any x,y∈V there exists a sequence

Definition 2.2(Weighted Graph)Given a graphandare two mappings on it.μ is symmetric on V.For convenience,we extend μ onto V×V,that is to say,for any x,y∈

Definition 2.3Let m be a measure defined as above.Then(V,m)is a measure space.We define anspace as follows:

Definition 2.8(Gradient Operator Γ2)The operator Γ2is defined as follows

These two lemmas are proved in[7].

Now we will introduce some basic CD inequalities(also see[4,8]).

Definition 2.11(CD(K,∞)Condition)We say that a graph satisfies CD(K,∞)condition if for any x∈V,we have

3 Main Results

When we look for the equivalent properties of CD inequalities,we often set a condition:And the equivalent properties were proved in[9]for these bounded Laplace operator on graphs.For the unbounded Laplace operator,the following equivalent properties under the condition of nondegenerate measure were proved in[7]by Bobo Hua and Yong Lin.

For f ∈ C0(V),from Lemma 2.1 and Theorem 2.1 we can easily getThen from Lemma 2.3,we get:

ForThen,the equation is changed into

Now we will show that G(s)is differentiable in s.

Without loss of generality,we assume that ?0.For the first part,from Lemma 2.4 we have

where C1is a constant satisfying

For the second part,from Lemma 2.4 we have

Obviously,this equation holds for all the finitely supported functions.

Now taking a series of functionsdefined as Definition 2.10.Letobviouslythen

[1]Chung,F.,Spectral Graph Theory,CBMS Regional Congerence Series in Mahtematics,92,American Mathemeatical Society,Providence,RI,1997.

[2]Hua,B.B.and Lin,Y.,Stochastic completeness for graphs with curvature dimension conditions,Adv.Math.,306,2017,279–302.

[3]Bakery,D.,Gentil,I.and Ledoux,M.,Analysis and geometry of Markov diffusion operators,Number 348 in Grundlehren der Mathematischen Wissenschaften.Springer,Cham,2014.

[4]Bauer,F.,Horn,P.,Lin,Y.,Lippner,G.,Mangoubi,D.and Yau,S.-T.,Li-Yau inequality on graphs,Journal of Differential Geometry,99,2015,359–405.

[5]Keller,M.and Lenz,D.,Unbounded Laplacians on graphs:basic spectral properties and the heat equation,Mathematical Modelling of Natural Phenomena,5(4),2011,198–224.

[6]Keller,M.and Lenz,D.,Dirichlet forms and stochastic completeness of graphs and subgraphs,Journal Fur Die Reine Und Angewandte Mathematik,2012(666),2012,1074–1076.

[7]Horn,P.,Lin,Y.,Liu,S.and Yau,S.-T.,Volume doubling,Poincar′e inequality and Guassian heat kernel estimate for nonnegative curvature graphs,preprint.

[8]Lin,Y.and Yau,S.-T.,Ricci curvature and eigenvalue estimate on locally finite graphs,Math.Res.Lett.,17(2),2010,343–356.

[9]Lin,Y.and Liu,S.,Equivalent properties of CD inequality on graphs,preprint.

[10]Yau,S.-T.and Schoen,R.,Lectures on Di ff erential Geometry,Higher Education Press,Beijing,2014.

[11]Gudmundsson,S.,An Introduction to Riemann Geometry,Beijing University,Beijing,2004.