LI-YAU INEQUALITY ON GRAPHS. Frank Bauer, Paul Horn, Yong Lin, Gabor Lippner, Dan Mangoubi, Shing-Tung Yau. Abstract

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1 LI-YAU INEQUALITY ON GRAPHS Frank Baer, Pal Horn, Yong Lin, Gabor Lippner, Dan Mangobi, Shing-Tng Ya Abstract We prove the Li-Ya gradient estimate for the heat kernel on graphs. The only assmption is a variant of the crvatredimension ineqality, which is prely local, and can be considered as a new notion of crvatre for graphs. We compte this crvatre for lattices and trees and conclde that it behaves more natrally than the already existing notions of crvatre. Moreover, we show that if a graph has non-negative crvatre then it has polynomial volme growth. We also derive Harnack ineqalities and heat kernel bonds from the gradient estimate, and show how it can be sed to derive a Bser-type ineqality relating the spectral gap and the Cheeger constant of a graph. 1. Introdction and main ideas In their celebrated work [19] Li and Ya proved an pper bond on the gradient of positive soltions of the heat eqation. In its simplest form, for an n-dimensional compact manifold with non-negative Ricci crvatre the Li-Ya gradient estimate states that a positive soltion of the heat eqation t = 0 satisfies 1.1 log t log = t n t. The ineqality 1.1 has been generalized to many important settings in geometric analysis. The most notable sch generalization was made by Hamilton on the Ricci flow, see [14, 15]. Finding a discrete version of 1.1 has proven challenging for a long time. Indeed, there is no graph for which ineqality 1.1 is tre for all times t cf The main difficlty for finding a discrete version is that the chain rle fails on graphs. In this paper, we scceed in finding an analoge of ineqality 1.1 on graphs. The main breakthrogh and novelty of this paper, as we see, are twofold. First, we show a way to bypass the chain rle in the discrete setting. The way we do it as explained in 1.1, we believe, may be adapted to many other circmstances. Second, we introdce a new natral notion of crvatre 1

2 BAUER, HORN, LIN, LIPPNER, MANGOUBI, YAU of graphs, modifying the crvatre notion of [0]. For example, we are able to prove a discrete version of the Li-Ya ineqality where the crvatre is bonded from below by any real nmber. Also, we show that non-negatively crved graphs have polynomial growth. As far as we are aware of, this reslt, well known on Riemannian manifolds, is not known with any previos notion of crvatre on graphs. In the next two sections we explain the preceding two ideas in more detail Bypassing the chain rle - discretizing the logarithm. In proving the gradient estimate 1.1 on manifolds, either by the maximm principle [19] or by semigrop methods [4] it is crcial to have the chain rle in hand. Namely, both proofs se a simple bt a key identity that follows from the chain rle formla: 1. log = log. However, this is false in the discrete setting. Even worse, there seems to be no way to reasonably bond the difference of the two sides. The lack of the chain rle on graphs is the main difficlty in trying to prove a discrete analoge of 1.1. In gerneral on graphs, the fact that the chain rle fails cases major problems. See also [7,, 8, 9] for other attempts to avoid or partially bypass the chain rle in the discrete setting. In what follows we explain or soltion to this isse. First, we consider a one parameter family of simple identities on manifolds which resembles 1., see also [7]: For every p > 0 one has 1.3 p = p p 1 + p 1 p p p. These also follow from the chain rle. Then, we make the following crcial observation: While there exists no chain rle in the discrete setting, qite remarkably, identity 1.3 for p = 1/ still holds on graphs. This fact is the starting point and probably the most important observation of this paper. Naively, this means that each time identity 1. is applied in the proof of 1.1 we may try to replace it by identity 1.3 with p = 1/. Indeed, this idea starts or work in this paper. However, this idea alone is not enogh to prove a discrete analoge of 1.1: We have to redefine the notion of crvatre on graphs as explained in the next section. 1.. A new notion of crvatre for graphs. The second obstacle we have to overcome in proving gradient estimates on graphs is that a proper notion of crvatre on graphs is not a priori clear. It is a well known problem to extend the notion of Ricci crvatre, or more precisely to define lower bonds for the Ricci crvatre in more general spaces than Riemannian manifolds. At present a lot of research has been done in this direction see e.g. [1, 0, 1, 4, 5, 8]. The approach

3 LI-YAU INEQUALITY ON GRAPHS 3 to generalizing crvatre in the context of gradient estimates by the se of crvatre-dimension ineqalities explained below was pioneered by Bakry and Emery [3]. On a Riemannian manifold M Bochner s identity reveals a connection between harmonic fnctions or, more generally, soltions of the heat eqation and the Ricci crvatre. It is given by f C 1 M f = f, f + Hessf + Ric f, f. An immediate conseqence of the Bochner identity is that on an n- dimensional manifold whose Ricci crvatre is bonded from below by K one has f f, f + 1 n f + K f, which is called the crvatre-dimension ineqality CD-ineqality. It was an important insight by Bakry and Emery [3] that one can se it as a sbstitte for the lower Ricci crvatre bond on spaces where a direct generalization of Ricci crvatre is not available. Since all known proofs of the Li-Ya gradient estimate exploit nonnegative crvatre condition throgh the CD-ineqality 1.4, one wold believe it is a natral choice in or case as well. Bakry and Ledox [4] scceed to se it to generalize 1.1 to Markov operators on general measre spaces when the operator satisfies a chain rle type formla. As we have explained in Section 1.1 there is no chain rle in the discrete setting. However, de to formla 1.3 with p = 1/ which compensates for the lack of the chain rle, we scceed to modify the standard CD-ineqality on graphs in order to define a new crvatre notion on graphs cf. 3 which we can se to prove a discrete gradient estimate in Theorem 4.4. One may arge that as we modify the crvatre notion it might not be natral anymore. In fact, we show it is natral in several respects: First, we prove that or modified CD-ineqality follows from the classical one in sitations where the chain rle does hold Theorem Second, we compte it in several examples 6 to show it gives reasonable reslts. In particlar, we show that trees can have negative crvatre K with K arbitrarily large. So far the existing notions of crvatre [0, 5] always gave K for trees. Third, as mentioned above, we derive polynomial volme growth for graphs satisfying non-negative crvatre condition, like on manifolds Corollary 7.8, and it seems to be a first reslt of this kind on graphs Backgrond on the parabolic Harnack ineqality on graphs. Ineqality 1.1 can be integrated over space-time, and some new distance fnction on space-time can be introdced to measre the ratio of the positive soltion at different points: 1.5 x, s Cx, y, s, ty, t,

4 4 BAUER, HORN, LIN, LIPPNER, MANGOUBI, YAU where Cx, y, s, t depends only on the distance of x, s and y, t in space-time. Using this, [19] also gave a sharp estimate of the heat kernel in terms of sch a distance fnction. Harnack ineqalities like 1.5 have many applications. Besides implying bonds on the heat kernel, they can be sed to prove eigenvale estimates, and it is one of the main techniqes in the reglarity theory of PDEs. Hence it is important to decide what manifolds satisfy sch an ineqality. Saloff-Coste [6] gave a complete characterization of sch manifolds see also Grigor yan [13] for an interesting alternative prove that the Poincaré ineqality in conjnction with volme dobling imples the Harnack ineqality. Saloff-Coste showed that satisfying a volme dobling property along with Poincaré ineqalities is actally eqivalent to satisfying the Harnack ineqality 1.5. The characterization by Saloff-Coste generalizes the non-negative Ricci crvatre condition by Li and Ya, since it is known by combining the Bishop-Gromov comparison theorem [5] and the work of Bser [6] that a lower bond on the Ricci crvatre implies volme dobling and the Poincaré ineqality see also the paper of Grigor yan, [13]. However, a major drawback of the characterization by Saloff-Coste is that showing that a manifold satisfies these properties is rather difficlt as both volme dobling and the Poincaré ineqality are global in natre. The reslts in [19] have the advantage that a simple local condition, a lower bond on crvatre, is sfficient to garantee that the more global properties hold, at the cost of being more sensitive to pertrbations. In the case of graphs Delmotte [10] proved a characterization analogos to that of Saloff-Coste for both continos and discrete time. However, jst as for manifolds, his conditions are hard to verify becase of their global natre. One virte of or reslts is that they give local conditions that imply Harnack type ineqalities. Organization of the paper. In Section we set p the scope of this paper. In Section 3 we define or notion of crvatre by modifying the standard crvatre-dimension ineqality, and we stdy the basic properties of the crvatre. In particlar, we show that on manifolds the modified CD-ineqality follows from the classical one. Or main reslts are contained in Section 4, where we establish the discrete analoge of the gradient estimate 1.1. In Section 5, we se the gradient estimates to derive Harnack ineqalities. Section 6 contains crvatre comptations for certain classes of graphs. In particlar we give a general lower bond for graphs with bonded degree and show that this bond is asymptotically sharp in the case of trees. We also show that lattices, and more generally Ricci-flat graphs in the sense of Chng and Ya [7], have non-negative crvatre. Finally, in Section 7 we apply or reslts

5 LI-YAU INEQUALITY ON GRAPHS 5 to derive heat kernel bonds, polynomial volme growth and prove a Bser-type eigenvale estimate.. Setp and notation First we fix or notation. Let G = V, E be a graph. We allow the edges on the graph to be weighted; that is, the edge xy from x to y has weight w xy > 0. We say that the graph is nweighted if w xy 1. We do not reqire that the edge weights be symmetric, so w xy w yx in general, for the proofs of the main theorems, bt or key examples satisfying the crvatre condition do have symmetric weights. We do, however, reqire that inf w e =: w min > 0. e E Moreover we assme in the following that the graph is locally finite, i.e. degx := w xy < for all x V. Given a measre µ : V R on V, the µ-laplacian on G is the operator : R V R V defined by fx = 1 µx w xy fy fx. Since sch averages will appear nmeros times in comptations, we introdce an abbreviated notation for averaged sm : For a vertex x V, hy := 1 w xy hy. µx Given a graph and a measre, we define D w = max x,y V x y degx w xy and degx D µ = max x V µx. So far as is possible, we will treat µ-laplacian operators generally. The special cases of most interest, however, are the cases where µ 1 which is the standard graph Laplacian, and the case where µx = w xy = degx, which yields the normalized graph Laplacian. Throghot the remainder of the paper, we will simply refer to the µ-laplacian as the Laplacian, except when it is important to emphasize the effect of the measre. In this paper, we are interested in fnctions : V [0, R that are soltions of the heat eqation. Let s introdce the operator L = t.

6 6 BAUER, HORN, LIN, LIPPNER, MANGOUBI, YAU We say that x, t is a positive soltion to the heat eqation, if > 0 and L = 0. It is not hard to see that sch soltions can be written as x, t = P t 0 when 0 =, 0 l p V, µ for some 1 p. Here P t = e t is the heat kernel and l p V, µ = {f : V R : fx p µx < }. Note that the heat eqation of corse also depends on the measre µ, throgh the Laplacian it contains. 3. Crvatre-dimension ineqalities In this section we introdce a new version of the CD-ineqality, which is one of the key steps in deriving analoges of the Li-Ya gradient estimate. We also compare or new notion to the standard CD-ineqality. First we need to recall [4] the definition of two natral bilinear forms associated to the Laplacian. Definition 3.1. The gradient form Γ = Γ is defined by Γf, gx = f g f g f g x = = 1 w xy fy fxgy gx. µx We write Γf = Γf, f. Similarly, the iterated gradient form Γ = Γ is defined by We write Γ f = Γ f, f. Γ f, g = Γf, g Γf, g Γ f, g. Definition 3.. We say that a graph G satisfies the CD-ineqality CDn, K if, for any fnction f Γ f 1 n f + KΓf. Note that this is exactly the CD-ineqality in 1.4 written in the Γ notation. G satisfies CD, K if Γ f KΓf. We shall drop the sperscript from the Γ and Γ notation nless it wold be confsing. Remark 3.3. The previos definitions can be applied to a broad class of operators, not jst or, and to a broad class of spaces, not jst graphs. In fact Bakry and Ledox [4] have shown that in a rather general setting CDn, 0 is sfficient to derive the gradient estimate 1.1 for the semigrop generated by an operator L, as long as it is a diffsion semigrop in the following sense

7 LI-YAU INEQUALITY ON GRAPHS 7 Definition 3.4. Given an operator L, the semigrop P t = e t L is said to be a diffsion semigrop if the following identities are satisfied for any smooth fnction Φ : R R Γ L f, g h = g Γ L f, h + h Γ L f, g Γ L Φ f, g = Φ fγ L f, g LΦ f = Φ flf + Φ fγ L f. The main example of sch a semigrop is the one generated by the Laplace-Beltrami operator on a Riemannian manifold. The Laplacian we are interested in does not generate a diffsion semigrop, bt remarkably, as we mentioned in the introdction, for the choice of Φf = f a key formla similar to a combination of 3.6 and 3.7 still holds: 3.8 f f = f Γ f. This can be viewed as the discrete analoge of 1.3 for p = 1/, and it is a simple conseqence of the identity a b a = b a b a. We haven t been able to find sefl discrete analoges of 1.3 for any other vale of p. The identity 3.8 motivates the following key modification of the CD-ineqality. Definition 3.9. We say that a graph G satisfies the exponential crvatre dimension ineqality at the point x V, CDEx, n, K if for any positive fnction f : V R sch that fx < 0 we have Γ fx Γ f, Γf f x 1 n fx + KΓfx. We say that CDEn, K is satisfied if CDEx, n, K is satisfied for all x V. Remark For convenience, we set 3.11 Γ f := Γ f Γ f, Γf. f or eqivalently by Γ f = 1 Γf Γ f, f f Remark An important aspect of both CDn, K and CDEn, K is that they are local properties: satisfying CDn, K or CDEn, K at any given vertex depends only on its radis neighborhood. Ths, in principle, it is possible to classify all nweighted.

8 8 BAUER, HORN, LIN, LIPPNER, MANGOUBI, YAU graphs which satisfy CDEn, K and have maximm degree at most D for any fixed D. Of corse, one hopes that typical graphs which one might consider to have non-negative crvatre satisfy CDEn, 0 for some dimension n. As we will show in Section 6, the class of Ricci-flat graphs [7], which incldes abelian Cayley graphs and most notably the standard lattices Z d along with finite tori do indeed satisfy CDEd, 0. Remark The reason we se the adjective exponential in Definition 3.9 is revealed in Lemma 3.15 below. Lemma If the semigrop generated by L is a diffsion semigrop, then for any positive fnction f one has Proof. We compte Γ f = f Γ log f. Γ log f =LΓlog f Γlog f, L log f Γf =L f f Γ f, Lf f Γf f = LΓf 1 1 f + Γ f, Γf + ΓfL f Γf, Lf f Lf f Γf, f 1 + Γf Γf, Γf + Γf, f f 3 f = Γ f f 4 Γf, Γf ΓfLf f 3 f 3 + f 3 LfΓf + f = Γ f f Γf, Γf + Γf f 3 f 4 = Γ f = f f 4 3 Γf, Γf 4Γf + 6 Γf f 4 f Γf, Γf Γf f 3 f Γf, f 1 Γ f Γ f, Γf = Γ f f f. q.e.d. Theorem If the semigrop generated by L is a diffsion semigrop, then the condition CDn, K implies CDEn, K.

9 LI-YAU INEQUALITY ON GRAPHS 9 Proof. Let f be a positive fnction sch that Lfx < 0. Lemma Γ f = f Γ log f f n L log f + K Γlog f = By On the other hand, 3.18 fxllog fx = Lfx Γfx fx Sqaring 3.18 and inserting the reslt in 3.17 yield Γ fx 1 n Lfx + KΓfx. = 1 n f L log f + KΓf. Lfx < 0. q.e.d. Remark Let s define that a graph satisfies the condition CDE n, K if for all f > 0, Γ f 1 n f log f + KΓf. In light of Lemma 3.15 and Theorem 3.16, for diffsion semigrops CDn, K CDE n, K CDEn, K, ths it is tempting to base or crvatre notion on CDE instead of CDE. Rather interestingly, making sch a definition in the graph case loses something: First, as we show below in Theorem 6.8, the integer grid Z d satisfies CDEd, 0. On the other hand, it only satisfies CDE 4.53d, 0 and this dimension constant essentially cannot be improved. Second, it trns ot that some graphs and, in particlar, reglar trees do not satisfy CDE n, K for any K > 0. In contrast, we show in Theorem 6.1 below that all graphs satisfy CDE, K for some K > Gradient estimates In this section we prove discrete analoges of the Li-Ya gradient estimate 1.1 for graphs satisfying the CDE-ineqality Failre of ineqality 1.1 on graphs. Let s start by explaining why 1.1 in its original form fails on graphs. Let G be any nontrivial graph and p, q V G two adjacent vertices. Let x, t be the soltion of the heat eqation with initial condition x, 0 = δ p. It is well-known that in this case x, t is the distribtion at time t of the continos time simple random walk started at p. The continos time random walk is obtained by having the walker iterate the following action: Wait for a random time according to an exponential distribtion and then

10 10 BAUER, HORN, LIN, LIPPNER, MANGOUBI, YAU take a random step. Let E 1, E be the exponential random variables sed in waiting to take the first two steps. Then for small t we have p, t P E 1 > t = 1 t 0 e s ds 1 t, q, t P E 1 < t t, 1 q, t degp P E 1 < tp E > t t1 t degp. Ths q, t 1 t q, t t > 1 t. On the other hand, t 0 > 0 small sch that t q, t 0 1. We conclde that t q, t 0 degp t 0 1 t 0 4 degp. t 0 The two terms together are still too big: q, t 0 q, t 0 t q, t 0 1 t 4 degp > n 0 t 0 t 0 Remark 4.1. There is a one-parameter family of possible gradient estimates: E p : p p Since in the Riemannian case p t n t. = p, the original Li-Ya ineqality corresponds to E 1, and the larger p is, the stronger E p is. p The argment preceding the remark in fact shows that E p cannot hold for any graph for any p > 0.5. The ineqalities we obtain in the seqel are for exactly p = 0.5 which, in this sense, is the best exponent one can hope for in discrete Li-Ya gradient estimates. 4.. Preliminaries. The following lemma, describing the behavior of a fnction near its local maximm, will be sed repeatedly throghot the whole section. Lemma 4.. Let GV, E be a finite or infinite graph, and let g, F : V [0, T ] R be fnctions. Sppose that g 0, and F has a local maximm at x, t V ]0, T ]. Then LgF x, t LgF x, t. Proof. On the one hand gf x, t = 1 µx w x ygy, t F y, t gx, t F x, t 1 µx w x ygy, t F x, t gx, t F x, t = gf x, t.

11 LI-YAU INEQUALITY ON GRAPHS 11 On the other hand t gf x, t = t gf x, t + g t F x, t t gf x, t, since t F = 0 at the local maximm if 0 < t < T and t F 0 if t = T. The last claim is jst the difference of the previos two. q.e.d. For convenience, we also record here some simple facts which we se repeatedly in or proofs of the gradient estimates. Lemma 4.3. Sppose f : V R satisfies f > 0, and fx < 0 at some vertex x. Then i fy fy < D µ fx. ii max w xy µx f y < D µ D w f x. Proof. i is obvios as f > 0. ii follows as f µx y fy < D µ D w f x. min w xy q.e.d Estimates on finite graphs. We begin by proving the gradient estimate in the compact case withot bondary. That is, we prove gradient estimates valid for positive soltions to parabolic eqations on finite graphs. Theorem 4.4. Let G be a finite graph satisfying CDEn, 0, and let be a positive soltion to the heat eqation on G. Then for all t > 0 Γ Proof. Let Γ 4.5 F = t t n t. t Fix an arbitrary T > 0. Or goal is to show that F x, T n for every x V. Let x, t be a maximm point of F in V [0, T ]. We may assme F x, t > 0. Hence t > 0. Moreover, by identity 3.8 which is tre both in the continos and the discrete setting, we know that Γ 4.6 F = t = t,.

12 1 BAUER, HORN, LIN, LIPPNER, MANGOUBI, YAU where we sed the fact that L = 0 recall that L = t which implies 4.7 t = t =. We conclde from 4.6 that 4.8 x, t < 0. In what follows all comptations are nderstood to take place at the point x, t. We apply Lemma 4. with the choice g =. This gives L F L F = Lt Γ = t LΓ Γ, where we sed 4.6 and the definition of L. We know that L = 0. Also, since and L commte, L = 0. So we are left with 4.9 F t = Γ t LΓ = = t Γ 4Γ, t = 4t Γ. The last eqality is tre by 3.1 and 4.7. By 4.8 and the CDEn, 0-ineqality applied to, t we get F t 4t 4.6 = t F n n t = nt F. Ths F n at x, t as desired. q.e.d. We can extend the reslt to the case of graphs satisfying CDEn, K for some K > 0 as follows. Theorem Let G be a finite graph satisfying CDEn, K for some K > 0 and let be a positive soltion to the heat eqation on G. Fix 0 < α < 1. Then for all t > 0 1 αγ t n 1 αt + Kn α. Proof. We proceed similarly to the previos case, so we do not repeat comptations that are exactly the same. Let F = t 1 αγ t. Fix an arbitrary T > 0, and we will prove the estimate at x, T for all x V. As before let x, t be the place where F assmes its maximm in the V [0, T ] domain. We may assme F x, t > 0 otherwise there is nothing to prove. Hence t > 0 and x, t < 0. In what follows all comptations are nderstood at the point x, t.

13 LI-YAU INEQUALITY ON GRAPHS 13 We again apply Lemma 4. with the choice g =. As before, this gives 0 = L F L F = Lt 1 αγ = = 41 αt Γ F t. Applying the CDEn, K ineqality to, mltiplying by t /1 α and rearranging give F 1 α 4t Γ CDEn, K = 1 n 4t 1 n KΓ = t 4t KΓ /. Let s denote G = t Γ /. Then F + αg = t ineqality can be rewritten as F 1 α 1 n F + αg t KG. so this After expanding F +αg we throw away the F G term, and se α G to bond the last term on the right hand side as follows. Completing the qadratic and linear term in G to a perfect sqare yields t 4.11 α G t Kn KnG = t Cα, n, K. α So we have F nf/1 α + t C, which implies F x, T F x, t n 1 α + t C n 1 α + T Kn α, which proves the gradient estimate at x, T for all x V. Since T is arbitrary, we have the theorem as claimed. q.e.d. We can also extend the reslt from soltions to the more general operator L q = t q = 0, where qx, t is a potential satisfying q ϑ and Γq η for some ϑ 0 and η 0. Theorem 4.1. Let G be a finite graph and qx, t : V R + R be a potential satisfying q ϑ and Γq η for all x V and t 0. Sppose = x, t satisfies L q = 0 on G. 1 If G satisfies CDEn, 0, then for all t > 0 Γ t q < n t + 1 nϑ + η D µ D w + 1.

14 14 BAUER, HORN, LIN, LIPPNER, MANGOUBI, YAU Fix 0 < α < 1, and 0 < ɛ < 1. If G satisfies CDEn, K, for some K 0, then for all t > 0 1 α Γ where Cα, n, K, ϑ, η, ɛ = t q < n 1 αt + 1 Cα, K, n, ϑ, η, ɛ, n 1 α ϑ + K n 1 ɛα + 4 n1 + αdw η 3. 1 αα 1/ ɛ 1/4 Proof. Again, the proof is qite similar to the proof of Theorem 4.4 so we do not repeat comptations that are exactly the same. Let Γ t F = t q As t q = 0, note t = q, so we may rewrite F as F = t Γ = t as before. Again, we fix an arbitrary T and take x, t to be the place where F assmes its maximm in the V [0, T ] domain, and we may assme that F x, t > 0 and hence t > 0 and x, t < 0. All comptations below shold be nderstood at the point x, t. We again apply Lemma 4. with the choice g =. The primary difference is that now at the maximm Then, similarly as before, 4.13 t q LF 4.14 LF LF = q = t q. = F t = F Rearranging 4.14, 4.15 Note F t t [ + t Γ Γ ] t, L + t 4 Γ + Γ, q q + t 4 Γ + Γ, q + q q q = q + q + Γ, q = q + q + Γ, q + Γ, q

15 LI-YAU INEQUALITY ON GRAPHS 15 Combining 4.15 and 4.16, we obtain 4.17 Finally, we bond 0 F t + t 4 Γ q Γ, q Γ, q Γ Γq < η D µ D w + 1. Here the first ineqality follows from an application of Cachy-Schwarz. The bond on Γ x, t follows as x, t < 0, and applying Lemma 4.3 ii yields With this, 4.17 gives Γ x, t = 0 > F t + t y, t x, t [y, t + x, t ] < D µ D w + 1 x, t. 4 Γ ϑ η D µ D w + 1. Applying the CDEn, 0 ineqality, mltiplying by nt / and rearranging yield F < nf t n ϑ + η D µ D w + 1 which yields the first claim of the theorem, as above. The general case with negative crvatre works by combining the above with the method of Theorem In order to get the best constant, a bit of additional care is needed, however. In the general case, 1 αγ t F = t q = t 1 α α Following the previos comptation, again at x, t maximizing F, 1 αt q αq LF + t 1 α = F t = F t [ Γ Γ, ] t L + t 41 α Γ + 1 αγ, q q.

16 16 BAUER, HORN, LIN, LIPPNER, MANGOUBI, YAU After some comptation and rearrangement, we get that > F t + t 41 α Γ 1 αϑ + Γ, q + αq q = F t + t 41 α Γ 1 αϑ + 1 α Γ, q α q + Γ, q. We handle the Γ, q term slightly differently here, simply applying Cachy-Schwarz to observe Γ, q Γ Γq η Γ. We similarly apply Cachy-Schwarz to bond Γ, q ΓΓq < η Γ. Lemma 4.3 and the fact that x < 0 to obtain that y < D w x for any y x. Combining these facts, we observe 4.19 Γ = x 1 y x y y = x 1 x 1 + x < D w + 1 x Γ. This establishes that q + Γ, q < ϑ + D w + 1η Γ. Following the comptations of the proof of Theorem 4.10 from 4.11; again setting G = t Γ /.. Applying the CDEn,-K ineqality, and mltiplying throgh by t /, we obtain from 4.18 the following: 0 > F + 1 α F + αg 1 αt KG n η1 + αd w t 3/ G t αθ Mltiplying ot, ignoring the positive F G term, and frther mltiplying the reslt by, we obtain n 1 α 0 > F n 1 α +F +α G t KnG η1+αd w t 3/ G t nθ 1 α.

17 LI-YAU INEQUALITY ON GRAPHS 17 A qartic in G arises, and for any ɛ 0, 1 we can write this qartic as 1 ɛα G t KnG + ɛα G η1 + αd w t 3/ G > t 1 αk n η1 + αdw 4/3 + ɛ ɛ 1/4 α 1/. 1 α In all, we obtain that for any ɛ 0, 1, where, Cα, n, K, ϑ, η, ɛ = F n 1 α > F t C α, n, K, ϑ, η, ɛ n 1 α ϑ + K n 1 ɛα + 4 n1 + αdw η 3, 1 αα 1/ ɛ 1/4 ths completing the reslt. Again, we prove the reslt for all x, T bt, as T is arbitrary, this completes the proof of the theorem. q.e.d General estimates in a ball. We can prove somewhat weaker reslts in the presence of a bondary. We do not assme finiteness of the graph anymore, and we only assme the heat eqation is satisfied in a finite ball. Or estimates will depend on the radis of this ball. We shall prove two types of estimates. In this section we prove the first type that works for any non-negatively crved graph, while the second type reqires the existence of so-called strong ct-off fnction on the graph that we will discss later in Section 4.5. Theorem 4.0. Let GV, E be a finite or infinite graph and R > 0, and fix x 0 V. 1 Let : V R R a positive fnction sch that Lx, t = 0 if dx, x 0 R. If G satisfies CDEn, 0 then for all t > 0 Γ t < n t + n1 + D wd µ R in the ball of radis R arond x 0. Let : V R R a positive fnction sch that L qx, t = 0 if dx, x 0 R, for some fnction qx, t so that q ϑ and Γq η. If G satisfies CDEn, K for some K > 0, then for any 0 < α < 1, any 0 < ɛ < 1 and all t > 0 1 αγ t q < < n 1 αt + n + D wd µ 1 αr + 1 Cα, n, K, ϑ, η, ɛ,

18 18 BAUER, HORN, LIN, LIPPNER, MANGOUBI, YAU where K n Cα, n, K, ϑ, η, ɛ = α + n 1 α ϑ+ [ + η 1 α D µ D w α D µ Dw in the ball of radis R arond x 0. ] 1/ Proof. First we consider the non-negative crvatre case. Let s define a ct-off fnction φ : V R as 0 : dv, x 0 > R R dv,x φv = 0 R : R dv, x 0 R 1 : R > dv, x 0 We are going to se the maximm-principle as in the proof of Theorem 4.4. Let F = tφ Γ = tφ, and let x, t be the place where F attains its maximm in V [0, T ] for some arbitrary bt fixed T. Or goal is to prove a bond on F x, T for all x V and as T is arbitrary this completes the proof. This bond is positive, so we may assme that F x, t > 0. In particlar this implies that t > 0, φx > 0, and x, t < 0. Let s first assme that φx = 1/R. Since positivity of implies that for any vertex x x = y 1 degx D µ, x µx we see that in this case F x, t t D µ /R and ths For x Bx 0, R, φ 1, so F x, T F x, t t D µ /R T D µ R. F x, T = T Γ x, T T D µ R, and dividing by T yields a stronger reslt than desired. We may therefore assme that φx R and φ does not vanish in the neighborhood of x. Now we apply Lemma 4. with the choice F = /φ. Ths we get L F L φ φ F = F t φ + t LΓ.

19 LI-YAU INEQUALITY ON GRAPHS 19 Using the fact that L = 0 we can write L = φ 1 φy 1 φx Using the same comptation as in 4.9 we get y. t LΓ = 4t Γ t n = t n F t φ Ptting these together and mltiplying throgh by t φ / we get φx t F 1 φy 1 y φx x + φf 1 n F.. Let s write φx = s/r. Then for any y x we have φy = s±1/r or φy = s/r. In any case 1 φy 1 φx R ss 1. Using Lemma 4.3 ii we have 1 φx t F φy 1 y φx x φx t F 1 φy 1 y φx x t F R y x < t D µ D w F. R Combining everything we can see that for any x sch that dx, x 0 R and ths φx = 1, at time T T Γ = F x, T F x, t < < n φ + nt deg x n + nt D wd µ, Rµxw min R and dividing by T gives the reslt. The proof of the general case is simply the combination of the preceding proof with that of Theorem 4.1. q.e.d. Corollary 4.1. If GV, E is an infinite, bonded degree graph satisfying CDEn, 0 and is a positive soltion to the heat eqation on G, then Γ t n t

20 0 BAUER, HORN, LIN, LIPPNER, MANGOUBI, YAU on the whole graph Strong ct-off fnctions. In the case of manifolds [19], a reslt similar to Theorem 4.0 holds with 1/R instead of 1/R. In one of the key steps of the argment the Laplacian comparison theorem is applied to the distance fnction. This together with the chain rle implies that one can find a ct-off fnction φ that satisfies φ cn 1 + R K R, where c is a constant that only depends on the dimension. Since the ct-off fnction φ also satisfies 4. φ φ < cn R it follows that there exists a constant Cn, that only depends on the dimension sch that 4.3 φ φ φ Cn 1 + R K R Unfortnately on graphs the Laplacian comparison theorem for the sal graph distance is not tre - think for instance of the lattice Z. This is the reason why in general we have to assme the existence of a ct-off fnction that has similar properties to 4. and 4.3, in order to prove a gradient estimate with 1/R. Noting that for a diffsion semigrop and hence in particlar for the Laplace-Beltrami operator on manifolds and φ 1 φ = φ + Γφ φ φ 3 Γ Cn1 + R K R 1 = Γφ φ φ Cn R this discssion motivates the following definition: Definition 4.4. Let GV, E be a graph satisfying CDEn, K for some K 0. We say that the fnction φ : V [0, 1] is an c, R-strong ct-off fnction centered at x 0 V and spported on a set S V if φx 0 = 1, φx = 0 if x S and for any vertex x S 1 either φx < c1+r K, R or φ does not vanish in the immediate neighborhood of x and φ x 1 φ x < D c1 + R K µ R and φ 3 xγ 1 φ x < D µ c R, where the constant c = cn only depends on the dimension n.

21 LI-YAU INEQUALITY ON GRAPHS 1 Remark 4.5. The strength of the strong ctoff fnction depends on the size of spport S. In order to get reslts akin to those in the manifold case, with 1 appearing for soltions valid in Bx R 0, cr, one reqires a strong ctoff fnction whose spport lies within a ball of radis cr. The ctoff fnction defined above, sing graph distance, gives a strong ctoff fnction on the ball of radis R. Theorem 4.6 yields a better estimate than Theorem 4.0 whenever one can find a strong ctoff fnction with spport in a ball of radis R. In Section 6 we will show see Corollary 6.10 and Proposition 6.14 that the sal Cayley graph of Z d with the reglar or the normalized Laplacian satisfies CDEd, 0 and admits a 100, R-strong ct-off fnction spported on a ball of radis dr centered at x 0. Theorem 4.6. Let GV, E be a finite or infinite graph satisfying CDEn, K for some K 0. Let R > 0 and fix x 0 V. Assme that G has a c, R-strong ct-off fnction spported on S V and centered at x 0. Fix 0 < α < 1. Let : V R R a positive fnction sch that L qx, t = 0 if x S, for some qx, t satisfying q ϑ and Γq η. Then for every ɛ 0, 1 1 αγ t n x 0, t < where + D µ cn 1 αr Cα, n, K, ϑ, η, ɛ = q 1 αt R K + nd w + 1 4α1 α n 1 α ϑ + K n 1 ɛα Cα, n, K, ϑ, η, ɛ, 4 n1 + αdw η 3. 1 αα 1/ ɛ 1/4 As we noted in the remark above, the lattice Z d yields a c, R-strong ctoff fnction in the ball Bx 0, dr and CDE0, d. As a reslt Theorem 4.6 specializes to the following. Corollary 4.7. If is a soltion of the heat eqation L = 0 in Bx 0, dr Z d, then with the choice α = 1/: Γ x 0, t 4d t + cd R for some explicit constant cd depending on the dimension. Proof of Theorem 4.6. We proceed similarly to the proof of Theorem 4.0, except that we assme φ is a c, R-strong ct-off fnction centered at x 0. Let s choose F = tφ 1 αγ,

22 BAUER, HORN, LIN, LIPPNER, MANGOUBI, YAU and let x, t denote the place where F attains its maximm in V [0, T ] for some arbitrary bt fixed T. Again, or goal is to show that F x, T is bonded for all x V, and since T is arbitrary this completes the reslt. We bond F by some positive qantity, hence we may assme F x, t > 0. This implies t > 0, φx > 0, and Γ 1 αγ > 0 at x, t. Hence x, t < 0 as in the proof of Theorem 4.0. First, if φx c1+r K then we are done, since R 1 αγ Γ = D µ, as we have seen in the proof of Theorem 4.0. Ths we may assme that Case of Definition 4.4 holds. In what follows all eqations are to be nderstood at x, t. We se Lemma 4. with the choice F = /φ to get 4.8 L F L φ φ F 4.9 = F t φ + t L1 αγ = F t φ + t [1 αlγ q ] = F t φ + t [ 41 α Γ + 1 αγ, ] q q. On the left hand side we se Cachy-Schwarz: L = L 1 1 φ φ + L + Γ φ φ, = q φ φ + Γ φ, 4.30 q φ + 1 φ + Γ 1 Γ, φ since L = q.

23 LI-YAU INEQUALITY ON GRAPHS 3 Collecting the q-terms in 4.9 and sing 4.30, we observe that they are t [ 1 αγ, q q ] q φ F = t [1 α Γ, q q q + + α q q] Γ Γ > t ϑ + 1 αη + αη Γ t ϑ + η1 + αd w In the comptation above we sed several times Cachy-Schwarz, 4.16 and the observation that Γ/ can be controlled by Γ / in the following way: By Lemma 4.3 i, and the fact that x < 0, we have that y < D w x for any y x. Hence Γ 4.33 = 1 y x = 1 y x 1 + < D w + 1 Γ. y x Combining 4.3 with 4.9 and mltiplying by t φ / we get 4.34 t φ ϑ + η1 + αd w + F t φ 3 Γ Γ + F t φ 1 φ + 1 φ Γ φ + φf > > 41 α Γ t φ. Let s introdce the notation G = t φγ /. Using 4.33, and that φ is a c, R-strong ct-off fnction we can frther estimate the left hand side of 4.34 from above: t φ ϑ+ η1 + αd w t φ 3 t D µ c1 + R K G + F + φf t D µ c D w + 1 R 1 R Γ F G > 41 α t φ.

24 4 BAUER, HORN, LIN, LIPPNER, MANGOUBI, YAU Using that the graph satisfies CDEn, K we can write 4t φ Γ 1 n t φ Kt φ Γ = Combining with 4.35 we have n t φ ϑ + η1 + αd w t φ 3 G 1 α + n 1 α = F + αg n t D µ c1 + R K R + φ + D w + 1 >F + αf G + α G t φkng t D µ c R t φkg. 1 G Notice that completing the left hand side to a perfect sqare gives αgf D w + 1 and hence n 1 α n 1 α n 1 α t D µ c R 1 GF t D µ c R D w + 1 n 4α1 α F t φ ϑ + η1 + αd w t φ 3 G t D µ c1 + R K R + φ + D w + 1 t D µ cn 4α1 αr > F + α G t φkng. Now we consider the terms from 4.37 containing G, namely α G t φkng n η1 + αadw t φ 3 G. 1 α We give a lower bond on these terms. Letting G = Γ = G/t φ we obtain for every ɛ 0, 1 t φ ɛα G + 1 ɛα G Kn G n η1 + αdw G 1 α t φ ɛα G K n 1 ɛα n η1 + αdw G 1 α t φ K n 4 n1 + 1 ɛα αdw η 3 1 αα 1/ ɛ 1/4, where the ineqalities follow from minimizing the involved sqare and qartic. F F

25 LI-YAU INEQUALITY ON GRAPHS 5 This combined with 4.37 now yields n t D µ c1 + R K 1 α R + φ + D w + 1 t D µ cn 4α1 αr [ + t φ n which easily implies F < where 1 α ϑ + K n n1 + 1 ɛα + αdw η 1 αα 1/ ɛ 1/4 F 4 ] 3 F, n t D µ c1 + R K 1 α R + φ + D w + 1 t D µ cn 4α1 αr + Cα, n, K, ϑ, η, ɛ = n 1 α ϑ + K n 1 ɛα + + t φcα, n, K, ϑ, η, ɛ 4 n1 + αdw η 3. 1 αα 1/ ɛ 1/4 Using that φ 1, φx 0 = 1, t T, and F x 0, T F x, t, and finally dividing by T we get the desired pper bond 1 αγ x 0, T < < n 1 α D µ c1 + R K R + 1 T + D w + 1 D µ cn 4α1 αr + + Cα, n, K, ϑ, η, ɛ. q.e.d. 5. Harnack ineqalities In this section we explain how the gradient estimates can be sed to derive Harnack-type ineqalities. The proof is based on the method sed by Li and Ya in [19], thogh the discrete space does pose some extra difficlties. In order to state the reslt in complete generality in particlar, when f is a soltion to L qf = 0 as opposed to a soltion to the heat eqation, we need to introdce a discrete analoge of the Agmon distance between two points x, and y which are connected in Bx 0, R. For a path p 0 p 1... p k define the length of the path to be lp = k. Then in a

26 6 BAUER, HORN, LIN, LIPPNER, MANGOUBI, YAU graph with bonded measre µ µ max : { µ max l P ϱ q,x0,r,µ max,w min,αx, y,, T = inf w min 1 αt + k 1 + k T ti+1 i=0 ti+1 t i qx i, tdt + t i } t t i qx i, t qx i+1, tdt where the infinm is taken over the set of all paths P = p 0 p 1 p p 3... p k so that p 0 = x, p k = y and having all p i Bx 0, R, and the times = t 0, t 1, t,..., t k = T evenly divide the interval [, T ]. In the case when the graph satisfies CDEn, 0 one can set α = 0. Remark 5.1. In the special case where q 0 and R =, which will arise when f is a soltion to the heat eqation on the entire graph, then ϱ simplifies drastically. In particlar, ϱ µmax,α,w min x, y, t 1, t = where dx, y denotes the sal graph distance. µ max dx, y 1 αt w min, Theorem 5.. Let GV, E be a graph with measre bond µ max, and sppose that a fnction f : V R R satisfies 1 α Γf f x, t tf f x, t qx, t c 1 t + c whenever x Bx 0, R for x 0 V along with some R 0, some 0 α < 1 and positive constants c 1, c. Then for < T and x, y V we have c1 T fx, fy, T exp c T + ϱ q,x0,r,µ max,w min,αx, y,, T In the case of nweighted graphs, and when dealing with positive soltions to the heat eqation everywhere, Theorem 5. simplifies greatly. Corollary 5.3. Sppose GV, E is a finite or infinite nweighted graph satisfying CDEn, 0, and µx = degx for all vertices x V. If is a positive soltion to the heat eqation on G, then n T 4Ddx, y x, y, T exp, T where D denotes the maximm degree of a vertex in G.

27 LI-YAU INEQUALITY ON GRAPHS 7 Remark 5.4. Observe that in the application of Theorem 5. to prove the corollary, one may take c 1 = n see Theorem 4.1 and Theorem 4.0, bt Theorem 5. natrally compares x, to x, T. To compare x, to x, T reqires sqaring both sides and introdces a factor of two in the exponent. Before we give the proof of Theorem 5., we need one simple lemma. Lemma 5.5. For any c > 0 and any fnctions ψ, q 1, q : [, T ] R, we have min ψs 1 s [,T ] c c T + T T s q 1 tdt + s ψ tdt + q 1 tdt + 1 T T T s q tdt t q t q 1 tdt. Proof. We bond the minimm by an averaged sm. Let φt = c t. Then min ψs 1 s [,T ] c φs ψs 1 c T c = T + c = T + T T T s q 1 t [ T T c T + T s s T ψ tdt + q 1 tdt + q tdt s ψ tdt + s q 1 tdt + T s T φsds φsψsds 1 c T t φsdsdt + T T t ψt ψ t c T t c T q 1 tdt + 1 T t ψ t q tdt ds φsdsdt t q t φsdsdt t T1 dt c T q 1 tdt + T t c ] q tdt t q t q 1 tdt. as we claimed, since x x 1. q.e.d. With this, we can retrn to the proof of Theorem 5..

28 8 BAUER, HORN, LIN, LIPPNER, MANGOUBI, YAU Proof of Theorem 5.. Let s first assme that x y. Then for any s [, T ] we can write log fx, log fy, T = log fx, fx, s fy, s + log + log fx, s fy, s fy, T s fx, s T = t log fx, tdt + log fy, s t log fy, tdt We se the assmption that to dedce t log f = tf f c 1 t + c 1 α Γf f + q T c 1 log fx, log fy, T t + s Γf + c dt 1 α x, tdt + f + s qx, tdt + T s T c 1 log T + c T 1 αw min + s µ max fx, s fy, s fy, s + s Γf fx, s f y, tdt + log fy, s qy, tdt T s s fy, t fx, t fy, t qx, tdt + T s + qy, tdt. In the second step we threw away the s term, and sed Γfy, t 1 w minfy, t fx, t /µ max as well as the fact that log r r 1 for any r R. We are free to choose the vale of s for which the right hand side is minimal. We se Lemma 5.5, with the choice ψt = fx, t/fy, t 1 and c = 1 αw min /µ max along with q 1 t = qx, t and q t = qy, t to get logfx, log fy, T µ max c 1 log T + c T + 1 αt w min T 1 T + qx, tdt + T t qy, t qx, tdt. To handle the case when x and y are not adjacent, simply let x = x 0, x 1,..., x k = y denote a path P between x and y entirely within Bx 0, R, and let = t 0 < t 1 < < t k = T denote a sbdivision of

29 LI-YAU INEQUALITY ON GRAPHS 9 the time interval [, T ] into k eqal parts. For any 0 i k 1 we can se 5.7 to get k 1 [ log fx, log fy, T = log fxi, t i log fx i+1, t i+1 ] k 1 i=0 + i=0 c 1 log t i+1 t i + c t i+1 t i + k T k 1 + i=0 ti+1 t i ti+1 t i qx i, tdt+ µ max 1 α T k w min t t i qx i, t qx i+1, tdt c 1 log T k µ max k 1 +c T αt w min + k T ti+1 Minimizing all paths, we have that log fx, log fy, T t i i=0 ti+1 t i + qx i, tdt t t i qx i, t qx i+1, tdt. Hence c 1 log T + c T + ϱ q,x0,r,µ max,w min,αx, y,, T. fx, fy, T as was claimed. T c1 expc T + ϱ q,x0,r,µ max,w min,αx, y, t 1, t 6. Examples q.e.d. In this section we show that or crvatre notion behaves somewhat as expected, by compting crvatre lower bonds for certain classes of graphs. We also show that Z d admits strong ct-off fnctions in the sense of Definition General graphs and trees. Here we prove that every graph satisfies CDE, D Dw µ + 1. We show that this bond is close to sharp for graphs that are locally trees, in particlar the crvatre of a D-reglar large girth graph goes to linearly as D.

30 30 BAUER, HORN, LIN, LIPPNER, MANGOUBI, YAU degx Theorem 6.1. Sppose G is any graph with D w = max x y w xy and D µ = max degx µx. Then G satisfies CDE, D Dw µ + 1 Proof. Fix a fnction f : V R with f > 0, and vertex x so that fx < 0. We begin by calclating: Γ fx = 1 [ Γf Γ f, f ] f [ = 1 Γfy Γfx 6. = 1 4 z y 1 f y fy fx f ] x fy fx [ fz fy fy fx f z f y fy 1 Γfx + 1 fy fx f x 4 fx = 1 [ ] fx 4 fy f z fyfz + f y fxfy z y 1 Γfx + 1 fx + Γf fx f, where in the second to last line we collected the terms at distance two, and in the last line we sed the identity that f x = fx fx+ Γfx. The smmands of the doble sm are qadratics in fz. They are minimized when fz = f y fy fx, whence the smmand is fx fx fy, so 6.3 Γ f 1 fy 4 fx fx fy z y 1 Γfx + 1 fx + Γf fx f 1 4 D fy µ fx fx fy 1 D µγfx + 1 fx + Γf fx f. ]

31 LI-YAU INEQUALITY ON GRAPHS 31 We se the fact that f = fy fx fx D µ fx, to lower bond the Γf fx f term. Finally, we se the fact that f < 0, and Lemma 4.3 i implies that Therefore, contining from 6.3, fy fx < D w. Γ f 1 4 D fy µ fx fx fy 1 D µγfx+ + 1 fx + Γf fx f > 1 Dw fx D µ + 1 Γf as desired. q.e.d. 6.. Sharpness of Theorem 6.1 on trees. For nweighted graphs with the normalized Laplacian, Theorem 6.1 states that all graphs satisfy CDE, D 1. Sch a lower bond on crvatre is essentially tight in the case of trees. Indeed, let T D, x 0 denote the infinite D-ary tree rooted at x 0. We find below fnctions f D for which Γ f D 6.4 Γf D 1 + o1d, as D. To constrct the fnction f D we do the following. Let y 1,..., y D denote the neighbors of x 0. We define fnctions f ɛ as follows: f ɛ x 0 = 1 f ɛ y 1 = 1 ɛd f ɛ y i = ɛ for i D. For vertices z y i at distance two from x 0, we take f ɛ z = f y i and hence, by the comptation in the proof of Theorem 6.1 being the vale that minimize Γ f ɛ given the f ɛ y i. Then we take f D = f ɛ for ɛ = D 3/. It is a straight forward comptation to verify that 6.4 holds Ricci-flat graphs. Chng and Ya [7] introdced the notion of Ricci-flat nweighted graphs as a generalization of Abelian Cayley graphs. Definition 6.5. A d-reglar graph GV, E is Ricci-flat at the vertex x V if there exist maps η i : V V ; i = 1,..., d that satisfy the following conditions.

32 3 BAUER, HORN, LIN, LIPPNER, MANGOUBI, YAU 1 xη i x E for every i = 1,..., d. η i x η j x for every i j. 3 for every i we have j η i η j x = j η j η i x In fact to test Ricci-flatness at x it is sfficient for the η i s to be defined only on x and the vertices adjacent to x. Finally, the graph G is Ricci-flat if it is Ricci-flat at every vertex. Given a weighted graph which is Ricci-flat when viewed as an nweighted graph, the weighting is called consistent if 1 There exist nmbers w 1,..., w d so that w xηi x = w i for all i = 1,..., d and x V. Whenever η j η i x = η i η k x for some x V then w j = w k. 3 The weights are symmetric, so w xy = w yx whenever x y. If only the first two conditions hold so the weights are not necessarily symmetric then we say the weighting is weakly consistent. Remark 6.6. The conditions on the weights are fairly restrictive, bt there are two cases when they are easily seen to be satisfied. 1 If w i = 1 : i = 1,..., d then we get back the original notion of Ricci-flat graph. If G is Ricci flat, and the fnctions η i locally commte, that is η i η j x = η j η i x, then any seqence w 1,..., w d can be sed to introdce a weakly consistent weighting for G. The critical reason why we choose these restrictions is the following: If G is a weakly consistently weighted Ricci-flat graph and f : V R is a fnction, then for any vertex x V, and 1 i d, 6.7 w j fη i η j x = w j fη j η i x. j j Here the fact G is Ricci flat implies the sms are over the same set of vertices, and the second condition on the weights ensres that the sms are eqal. Theorem 6.8. Let G be a d-reglar Ricci-flat graph. Sppose that the measre µ defining satisfies µx µ for all vertices x G. 1 If the weighting of G is consistent, then G satisfies CDEd, 0. If the weighting of G is weakly consistent, then G satisfies CDE, 0 Remark 6.9. For a d-reglar Ricci-flat graph and a weakly consistent weighting the two standard choices of the measre µ 1 and µx = degx satisfy µx µ for all x V. Corollary The sal Cayley graph of Z k satisfies CDEk, 0, for the reglar or normalized graph Laplacian.

33 LI-YAU INEQUALITY ON GRAPHS 33 Proof of Theorem 6.8. Let f : V R be a fnction. We begin by assming that G is Ricci flat, and the weighting is weakly consistent. We will write y for fx, y i for fη i x, and y ij for fη j η i x. With this notation we have Γfx = 1 w i Γfη i x Γfx µ and i = 1 µ w i w j yij y i y j y i j = 1 µ w i w j yij yj + yi y y i y ij + yy j, i,j Γ f, f = f = 1 y ij yi µ w i w j y i y y j y y i j i y = 1 y ji yi µ w i w j y i y y j y y i j i y = 1 y ij yj µ w i w j y j y y i y y i j j y = 1 µ w i w j yij yj + yi y + yy j y yij + y i y j. yy j i j Here, the second eqality follows from the weakly consistent labeling as observed in 6.7 and the third eqality follows from changing the role of i and j. Combining, we see Γ f = 1 Γx Γ f, f f = 1 y yij 4µ w i w y iy j y ij y + yi y j j yy j 6.11 ij = 1 yy ij y i y j 4µ w i w j. yy j ij Clearly, Γ f 0, so G satisfies CDE, 0 proving the first part of the assertion.

34 34 BAUER, HORN, LIN, LIPPNER, MANGOUBI, YAU Now we frther assme that the weighting of G is consistent. That is, we frther assme the weights are symmetric. Now for each i there is a niqe j = ji sch that η j η i x = x and ths y ij = y. Throwing away all the other terms from 6.11 we get: Γ 1 y y i y ji 4µ w i w ji. yy i i Note that ji is a fll permtation, and the symmetry of weights implies that w i = w ji, and hence on the cycles in ji the weights are constant. Sppose the permtation ji decomposes into cycles C 1,..., C k, with lengths l 1,..., l k. We focs or attention on an arbitrary cycle C. Then there exists a w C, and the terms above corresponding to this cycle are of the form wc y y i y ji 4µ yy i i C = w C y 4µ i C 1 z i z ji z i = = w C y 4µ i C 1 z i z ji + z i z ji where we take z i = y i /y. We can assme withot loss of generality that ji restricted to this cycle C is a permtation on [l], and 0 < z 1 z z l. We can apply the Rearrangement Ineqality to obtain zi zji z i zl+1 i and hence 6.1 w C y 4µ i C 1 z i z l+1 i + z i z l+1 i 1 8µ 1 z i z l+1 i + 1 z i = w C y i C w C y 4µ w C y i C 1 z i z l+1 i zi z l+1 i µ 1 z i z l+1 i i C = 1 µ w C y y i y l+1 i. i C z l+1 i We now combine the cycles together and apply Cachy-Schwarz, to see 6.13 Γ f 1 1 w i y y i y i, d µ where y i is the partner of y i in its cycle as given in 6.1. i,

35 LI-YAU INEQUALITY ON GRAPHS 35 Finally, we assme that fx < 0 to prove CDE conditions. This implies that i w iy i < i w iy. Also from the fact that y i and y i appear in the same cycle, we have i w iy i = i w iy i. Applying Cachy- Schwarz we see that w i yi y i w i y i w i y i = w i y i < w i y. i i i i i Ths contining 6.13, we see the interior sqare is positive, and hence Γ f 1 d as desired. 1 µ w i y y i y i i 1 d 6.4. Strong ct-off fnction in Z d. 1 µ w i y y i = 1 d f i q.e.d. Proposition The sal Cayley graph of Z d, along with a strongly consistent weighting, admits a 100, R-strong ct-off fnction spported in a ball of radis dr centered at the origin. Remark In the case of the Cayley graph of Z d, a strongly consistent weighting jst means that for each of the d generators e i, w xei x = w xe 1 i x. We did not attempt to optimize the constant 100 appearing in this statement. Proof. For a vertex x Z d let x i Z denote its ith coordinate and write x = i x i. We are going to prove that the fnction φx = max {0, R x } is a 100, R-strong ct-off fnction centered at the origin. It is spported in a Eclidean ball of radis R which is contained in a ball of radis dr measred in the graph distance. We need to show that one of the two cases in Definition 4.4 are satisfied. If R x 10R then the first case is clearly satisfied, so we may assme R x > 10R. Also, x i < R for any i, otherwise φx wold be 0. These together imply that 6.16 R R x R x ± x i x i 1 R x 1 1 3R 10R 10 7.

36 36 BAUER, HORN, LIN, LIPPNER, MANGOUBI, YAU By the consistency, for each coordinate there is a single weight w i. Now we can compte µxφ x 1 φ x = R x 4 R 4 = R 1 w i R x x i i R x + x i 1 R x R x = w i R R x R x 1 + 4x i R x 1 4x i R x x i 1 R x + x i 1 1 1x i R 4 w R x 4 + R x 5 i R x x i 1 R x + x i 1. i In the last line, we sed that R x 1 R x and discarded some negative terms. Then sing 6.16 along with x i < R and R x < R, we have φ x 1 φ x 1 µx i i w i 10/74 R < 100 R D µ. A comptation similar in spirit, bt less complicated, shows that 1 φ 3 xγ x = R x 6 φ R 1 µx R 4 w i R x R 4 R x ± x i R D µ, i and ths φ indeed is a 100, R-strong ct-off fnction. 7. Applications q.e.d Heat Kernel Estimates and Volme Growth. One of the fndamental applications of the Li-Ya ineqality, and more generally parabolic Harnack ineqalities, is the derivation of heat kernel estimates. As allded to in the introdction, Grigor yan and Saloff-Coste in the manifold setting and Delmotte in the graph setting proved the eqivalence of several conditions inclding Harnack ineqalities, and the combination of volme dobling and the Poincaré ineqality to the heat kernel satisfying the following Gassian type bonds. Let P t x, y denote the fndamental soltion to the heat eqation starting at x.