Curvatures, graph products and Ricci flatness

In this paper, we compare Ollivier–Ricci curvature and Bakry–Émery curvature notions on combinatorial graphs and discuss connections to various types of Ricci flatness. We show that nonnegativity of Ollivier–Ricci curvature implies the nonnegativity of Bakry–Émery curvature under triangle‐freeness and an additional in‐degree condition. We also provide examples that both conditions of this result are necessary. We investigate relations to graph products and show that Ricci flatness is preserved under all natural products. While nonnegativity of both curvatures is preserved under Cartesian products, we show that in the case of strong products, nonnegativity of Ollivier–Ricci curvature is only preserved for horizontal and vertical edges. We also prove that all distance‐regular graphs of girth 4 attain their maximal possible curvature values.

1. Introduction 1.1.Motivation of the paper.Curvature is a fundamental notion in the setting of smooth Riemannian manifolds.There is no unique choice of an analogue of curvature in the setting of combinatorial graphs.Two possibilities are Ollivier Ricci curvature and Bakry-Émery curvature which are both motivated by specific curvature properties of Riemannian manifolds.Ollivier Ricci curvature, introduced in [16], is based on the observation that, in the case of positive/negative Ricci curvature, average distances between corresponding point in two nearby small balls in Riemannian manifolds are smaller/larger than the distance between their centres.This fact is reinterpreted using the theory of Optimal Transportation of probability measures representing these balls.Bakry-Émery curvature, introduced in [1], is based on the socalled curvature-dimension inequality which reads for n-dimensional Riemannian manifolds (M, g) as follows: (1) 1 2 ∆ gradf 2 (x) ≥ ∇f (x), ∇∆f (x) + 1 n (∆f (x)) 2 + Ric(∇f, ∇f )(x) for all f ∈ C ∞ (M ) and x ∈ M .Here, Ric(v, w) for tangent vectors v, w at x stands for the Ricci curvature of the manifold.This formula is a straightforward implication of Bochner's identity, a fundamental fact in Riemannian Geometry with many important consequences.Both curvature notions have been further discussed in the setting of graphs in several literatures (see, e.g., [14] for Ollivier Ricci curvature and [11,15,18] for Bakry-Émery curvature).For the precise definitions of both notions in this paper, we refer to Section 2. While there are many special cases in which these two discrete curvature notions are related, it is a challenging problem to develop a satisfactory general understanding of the agreements and differences of these two curvature notions.
One special family of graphs which have both non-negative Ollivier Ricci curvature and non-negative Bakry-Émery curvature was introduced by F.R.K. Chung and S.-T.Yau [6], namely Ricci flat graphs.The notion of Ricci flatness was motivated by the structure of the ddimensional grid Z d (with vanishing Ollivier Ricci and Bakry-Émery curvature) and the class of Ricci flat graphs contains all abelian Cayley graphs as a subset.
The motivation of this paper is to investigate various relations between these two curvature notions and the property of Ricci flatness with special focus on triangle-free graphs.We also present explicit examples of graphs related to our results.The curvatures of these examples were calculated numerically via the interactive web-application at https://www.mas.ncl.ac.uk/graph-curvature/For more details about this very useful tool we refer the readers to [8].
The Bakry-Émery curvature is defined on vertices and the above inequality (1) involves a dimension parameter n.Since graphs do not have a well-defined dimension, a natural choice simplifying this inequality is n = ∞.The induced Bakry-Émery curvature value at a vertex x is then denoted by K ∞ (x) (see Definition 2.8).
we also introduce stronger types of Ricci flatness, namely (R)-, (S)and (RS)-Ricci flatness (see Definition 3.1 below).A fundamental consequence of Ricci flatness is that it implies both non-negativity of Ollivier Ricci and Bakry-Émery curvatures; the stronger property of (R)-Ricci flatness implies even strict positivity of these two curvatures (see Theorems 3.4 and 3.5).
Another basic property of Ricci flatness is that it is preserved under natural graph products (see Theorem 5.2).The graph products under consideration namely, Cartesian product (involving horizontal and vertical edges), tensorial product (involving only diagonal edges), and the strong product (involving all three types of edges), are introduced in Definition 5.1 below.While Cartesian products preserve non-negativity of both Ollivier Ricci curvature and Bakry-Émery curvature, in the case of strong products, non-negative Ollivier Ricci curvature is only preserved for horizontal and vertical edges (see Corollary 5.4).
We also consider the case of graphs which contain no triangles.In Section 4, we present our main result of this paper relating the two curvature notions.P. Ralli [17] gave an interesting criterion for curvature sign agreement of both curvature notions for triangle-free graphs which do not contain the complete bipartite graph K 2,3 as a subgraph.He mentions that the situation is much more unclear if one restricts to general triangle-free graphs.Our result requires triangle-freeness at a vertex x and the additional assumption that the in-degrees of vertices in the 2-sphere S 2 (x) are smaller or equal to 2. This assumption is weaker than non-existence of K 2,3 as a subgraph.
Theorem 1.2.Given a regular graph G = (V, E), let x ∈ V be a vertex not contained in a triangle and satisfying d − x (z) ≤ 2 for all z ∈ S 2 (x).Then we have the following: It is an important remark here that κ 0 (x, y) = 0, κ LLY (x, y) = 2 d , and K ∞ (x) = 2 are the maximum possible values of curvature for a vertex x not contained in a triangle.This curvature comparison result is proved by employing Ricci flatness, see Section 4. At the end of the section, we provide also examples to show that all conditions of the theorem are necessary.
In the final Section 6, we show that the curvatures of all distanceregular graphs of girth 4 and vertex degree d satisfy κ 0 = 0, κ LLY = 2 d and K ∞ = 2 (see Theorem 6.2).In other words, all curvatures attain their maximal possible values for this interesting family of triangle-free graphs.

Curvature notions
All graphs G = (V, E) with vertex set V and edge set E in this paper are simple (that is, without loops and multiple edges), undirected and connected, and we assume that the vertex degrees d x of all vertices x ∈ V are finite.Moreover, all our graphs are regular (that is d x = d for all x ∈ V ) unless stated otherwise.Balls and spheres are denoted by 2.1.Ollivier Ricci curvature.We define the following probability distributions µ p x for any x ∈ V, p ∈ [0, 1]: Definition 2.1 (Transport plan and Wasserstein distance).
The cost of a transport plan π is given by The set of all transport plans satisfying (2) is denoted by Π(µ 1 , µ 2 ).
In other words, a transport plan π moves a mass distribution given by µ 1 into a mass distribution given by µ 2 , and W 1 (µ 1 , µ 2 ) is a measure for the minimal effort which is required for such a transition.
The Ollivier Ricci curvature introduced by Lin, Lu, and Yau [14], is defined as It was shown in [14, Lemma 2.1] that the function p → κ p (x, y) is concave, which implies (4) Moreover, we have the following relation for edges {x, y} with d x = d y = d (see [3]): (5) From the definition of the Wasserstein metric we can get an upper bound for W 1 by choosing a suitable transport plan.Using Kantorovich duality (see e.g.[20,Ch. 5]), a fundamental concept in the optimal transport theory, we can approximate the opposite direction: Theorem 2.4 (Kantorovich duality).Given G = (V, E), let µ 1 , µ 2 be two probability measures on V .Then where 1-Lip denotes the set of all 1-Lipschitz functions.If φ ∈ 1-Lip attains the supremum we call it an optimal Kantorovich potential transporting µ 1 to µ 2 .
Note that both curvatures κ 0 (x, y) and κ LLY (x, y) of an edge {x, y} are already determined by the combinatorial structure of the induced subgraph B 2 (x).(In fact, by symmetry reasons, the combinatorial structure of the induced subgraph B 2 (x) ∩ B 2 (y) is sufficient.) As the relation κ 0 ≤ κ LLY is known from (4), now we will prove the surprising fact that strict positivity of κ LLY implies non-negativity of κ 0 (as stated in Theorem 1.1 from the Introduction).
Proof of Theorem 1.1.Let G = (V, E) be d-regular.Using the relation (5), it suffices to prove Let {x, y} ∈ E be an edge with κ 1 d+1 (x, y) > 0. We define the following sets: In other words, T xy is the set of common neighbours of x and y, V x is the set of neighbours of x which have distance 2 to y and, similarly, V y is the set of neighbours of y which have distance 2 to x.
We can choose an optimal transport plan π opt ∈ Π(µ x , µ The existence of an optimal transport plan satisfying (ii) (that is, without splitting mass), follows from [4, Theorem 1.1] (see also [19, p. 5]).Moreover, this transport plan can be chosen to satisfy (i) by [3,Lemma 4.1].Note that (iii) holds for any transport plan in Π(µ In other words, the optimal transport plan does not move the mass distributions at x, y or T xy , and for the vertices in V x it moves the mass distribution from one vertex completely to one vertex in V y .Thus the optimal transport plan pairs the vertices at V x and V y .Let u ∈ V x and denote by ũ the unique vertex in V y for which π opt (u, ũ) = 1 d+1 .Let us then consider the Wasserstein distance.Using the optimal transport plan we can write (6) 1 x , µ Now we distinguish three cases.
Next, we assume N 3 = 0 and N 2 > 0. Then there exists at least one vertex w ∈ V x satisfying d(w, w) = 2, and we obtain, similarly as above, and therefore κ 0 (x, y) ≥ 0.
The examples (b) and (c) show that the result in the theorem is sharp.
We finish this subsection with the following upper curvature bounds for κ 0 and κ LLY : Theorem 2.6 (see [13,Theorem 4] and [7,Proposition 2.7]).Let G = (V, E) be d-regular and {x, y} ∈ E. Then and where # ∆ (x, y) is the number of triangles containing {x, y}.
2.2.Bakry-Émery curvature.This curvature notion was first introduced by Bakry and Émery in [1] and was applied on graphs in [11,15,18].The definition of this curvature is based on the curvaturedimension inequality (1), which is equivalently rewritten as (8) below with the help of the following Γ-calculus.
For any function f : V → R and any vertex x ∈ V , the (nonnormalized) Laplacian ∆ is defined via ∆f (x) := We call K a lower Ricci curvature bound of x, and N a dimension parameter.The graph G = (V, E) satisfies CD(K, N ) (globally), if all its vertices satisfy CD(K, N ).At a vertex x ∈ V , let K(x, N ) be the largest K such that (8) holds for all functions f at x for a given N .We call K(x, •) the Bakry-Émery curvature function of x and we define In this paper, we will restrict our considerations to the curvature at ∞-dimension K ∞ : V → R. Note that for the definition of K ∞ (x), the formula (8) simplifies to where f , g are the vector representations of f and g.The matrices Γ(x), Γ 2 (x) are symmetric with non-zero entries only in B 1 (x) and B 2 (x), respectively.So we can view them as local matrices by disregarding the vertices outside B 2 (x).For the explicit matrix entries of Γ(x) and Γ 2 (x) see [9, Subsections 2.2 and 2.3].Note that these entries are already fully determined by the combinatorial structure of the incomplete 2-ball around x, denoted by B inc 2 (x), which is the induced subgraph of B 2 (x) with all edges within S 2 (x) removed.
We have the following general upper curvature bound similar to Theorem 2.6: Theorem 2.9 (see [9,Corollary 3.3]).Let G = (V, E) be d-regular and x ∈ V .Then where # ∆ (x) is the number of triangles containing x.
Let us finally return to the examples from the previous subsection.
Remark 2.10.The examples in Remark 2.5 have the following Bakry-Émery and Ollivier Ricci curvatures: None of the regular graphs in the above table have curvature with opposite signs.We are not aware of any such examples and it would be interesting to find such graphs.

Ricci flat graphs
The notion of Ricci flat graphs was introduced in 1996 by Chung and Yau [6] in connection to a logarithmic Harnack inequality and is motivated by the structure of the d-dimensional grid Z d .Abelian Cayley graphs are prominent examples of Ricci flat graphs.
) for all i.We also consider the following additional properties of the maps η i : (R) Reflexivity: η 2 i (x) = x for all i, (S) Symmetry: η j (η i x) = η i (η j x) for all i, j.If there exists a family of maps η i for a given vertex x ∈ V satisfying property (R) or property (S) in addition to (i)-(iii), we say that x is (R)-Ricci flat or (S)-Ricci flat, respectively.If there exists a family of maps η i satisfying (i)-(iii) and (R) and (S) simultaneously, we say that x is (RS)-Ricci flat.
The following lemma is a useful observation for the study of Ricci flatness of concrete examples.Lemma 3.2.Assume a family of maps η i : B 1 (x) → V satisfies (i)-(iii) of the above definition.Then each of these maps η i is a bijective map between B 1 (x) and B 1 (η i x).
Proof.Assume that the family η i satisfies (i)-(iii).It follows immediately from (i) and (ii) and regularity that This implies that (iii) is equivalent to Therefore, each map η i must be injective, since Bijectivity from B 1 (x) to B 1 (ηx) follows immediately from (9).
Note that all Ricci flatness properties at a vertex x can be determined from the combinatorial structure of the incomplete 2-ball B inc 2 (x) around x, which was introduced in Subsection 2.2.
Example 3.3.To help readers familiarize with the notion of Ricci flatness, we provide three examples of graphs and check whether each of them is Ricci flat.
(a) The incomplete 2-ball in Figure 2 with We show this by contradiction.Assume This implies that we have the following choices for our maps η j : x, v 5 , v 6 Such a table can be presented concisely with the help of a d × d matrix A, namely, A = (A ij ) defined as follows: Let S 1 (x) = {v 1 , . . ., v d } where v j := η j (x), and S 2 (x) =: {v d+1 , . . ., v t } and, furthermore, v 0 := x.Then the entries A ij ∈ {0, 1, . . ., t} of A are given via the relation Then the table translates into the following possibilities for the entries of A: The conditions (i)-(iii) require that all columns and rows of A have non repeating entries.Obviously, this is not possible in this case.Henceforth, we will use this matrix notation to simplify matters.(c) Shrikhande graph: Cayley graph Z 4 × Z 4 with the generator set {±(0, 1), ±(1, 0), ±(1, 1)}.It is a strongly regular graph (see [5, pp. 125]).The structure of the incomplete 2-ball B inc 2 (x) around any vertex x is given in Figure 3.We have the following possibilities for the entries of the associated matrix A: Choosing 0 for diagonal entries fixes all other entries of the matrix.Moreover, this choice leads to a symmetric matrix, which shows that x is (RS)-Ricci flat.

Ricci flatness and Ollivier Ricci curvature. With regards to
Ollivier Ricci curvature we have the following general implications: Proof.For the proof of (a) we assume Ricci flatness at x with corresponding maps Therefore, we have y = η i (x) for some i ∈ {1, . . ., d}.We choose the following transport plan: and π(u, v) = 0 for all other combinations.This implies and (using Lemma 3.2) which implies κ 0 (x, y) ≥ 0. We prove (b) similarly.Assume x is (R)-Ricci flat with corresponding maps η i and y = η i (x).Note that we have η i (y) = x from reflexivity.This time, we choose the following transport plan π ∈ Π(µ , and π(u, v) = 0 for all other combinations.This leads to

3.2.
Ricci flatness and Bakry-Émery curvature.With regards to Bakry-Émery curvature we have the following general implications: Proof.The proof of statement (a) was already explained in [6] and [15].This proof stategy can also be applied to prove statement (b).
We present these proofs for the reader's convenience.
Recall from the definition that A useful identity to compute Γ(f, g) is Let us now consider the first term on the RHS in (10) and use the identity On the other hand, we have for the second term on the RHS of (10), using Ricci flatness, Adding both terms, we end up with showing K ∞ (x) ≥ 0. Under the stronger condition of (R)-Ricci flatness, we can estimate 2Γ 2 (f, f )(x) from below as follows: This shows that Γ 2 (f, f )(x) ≥ 2Γ(f, f )(x), which means that we have

Triangle-free graphs
In this section we focus on curvature comparison results for graphs without triangles.Our main result states that non-negativity of Ollivier Ricci curvature implies non-negativity of Bakry-Émery curvature under a certain in-degree condition (see Corollary 1.2).This result is derived via Ricci flatness properties.
We start with particular upper curvature bounds in case of trianglefreeness: Proposition 4.1.Let G = (V, E) be d-regular.Then we have the following upper curvature bounds: (i) κ 0 (x, y) ≤ 0 for all edges {x, y} ∈ E not contained in a triangle, (ii) κ LLY (x, y) ≤ 2 d for all edges {x, y} ∈ E not contained in a triangle, (iii) K ∞ (x) ≤ 2 for all x ∈ V not contained in a triangle.Remark 4.2.Combining the proposition with the lower curvature bounds for Ricci flatness (Theorems 3.4 and 3.5), we obtain the following curvature equalities: • If x is Ricci flat and the egde {x, y} ∈ E is not contained in any triangle then κ 0 (x, y) = 0.
Proof of Proposition 4.1.Although Statements (i) and (ii) are an implication from Theorem 2.6, we provide their proof here which presents a useful idea for the following remark.
Statement (i) follows from since S 1 (x) ∩ S 1 (y) = ∅.Here π opt is an optimal transport plan in Π(µ 0 x , µ 0 y ).For the proof of (ii), we only need to show . This follows from Here π opt is an optimal transport plan in Π(µ Statement (iii) is an implication from Theorem 2.9.
Remark 4.3.Note that in Proposition 4.1, (ii) implies (i) by Theorem 1.1.Moreover, it follows from the above proof that sharpness of the bounds in (i) and (ii) has the following combinatorial interpretation in the triangle-free case: (a) κ 0 (x, y) = 0 is equivalent that there is a perfect matching between S 1 (x) and S 1 (y).
A natural class of examples where all three upper bounds of Proposition 4.1 are attained are distance-regular graphs of girth 4 (see Section 6 below).To motivate our next result, let us focus on one particular example: In fact this is the 2-ball of the d-dimensional hypercube Q d and we have the following curvatures (see Remark 2.10): We also like to mention that the vertex x in this example is (RS)-Ricci flat and that we have d − x (z) = 2 for all z ∈ S 2 (x).Theorem 4.5.Given a regular graph G = (V, E), let x ∈ V be a vertex not contained in a triangle and satisfying d − x (z) ≤ 2 for all z ∈ S 2 (x).Then we have the following: (a) κ 0 (x, y) = 0 for all y ∈ S 1 (x) is equivalent to x being (S)-Ricci flat.
This result, together with Theorem 3.5, implies our main curvature comparison result in Theorem 1.2 from the Introduction: Proof of Theorem 1.2.Under the assumptions of Theorem 4.5, we first assume that κ 0 (x, y) = 0 for all y ∈ S 1 (x).This implies that x is Ricci flat and, by Theorem 3.5(a), that K ∞ (x) ≥ 0.
Similarly, assuming κ LLY (x, y) = 2 d for all y ∈ S 1 (x), we know that x is (R)-Ricci flat, and Theorem 3.5(b) implies that K ∞ (x) ≥ 2. Since x is not contained in a triangle, this leads to K ∞ (x) = 2 by Proposition 4.1(iii).
Before we start with the proof of Theorem 4.5, let us introduce the following notion and discuss relations to existing results.Definition 4.6.Let G = (V, E) be a regular triangle-free graph and x ∈ V .We say that y 1 , y 2 ∈ S 1 (x) are linked by z ∈ S 2 (x) if we have y 1 ∼ z ∼ y 2 .We refer to z as a link of y 1 and y 2 .P. Ralli [17] investigated curvature implications for regular graphs without K 3 and K 3,2 as subgraphs.It is easy to check that this condition is equivalent to the following properties at all vertices x: (i) x is not contained in a triangle, (ii) d − x (z) ≤ 2 for all z ∈ S 2 (x), (iii) Any pair y 1 , y 2 ∈ S 1 (x) has at most one link.
A consequence of his results is that conditions (i),(ii),(iii) imply K ∞ (x) ≤ 0 or K ∞ (x) = 2.Under these conditions, Ralli has the following equivalence: κ 0 (x, y) = 0 for all y ∈ S 1 (x) ⇐⇒ K ∞ (x) ≥ 0. Our theorem implies that the implication "=⇒" holds already under conditions (i) and (ii) and we have an example that the implication "⇐=" is no longer true if one drops condition (iii).Let us now prove the forward implication in (a).Let x ∈ V be given with d = d x and S 1 (x) = {y 1 , . . ., y d }.The property κ 0 (x, y) = 0 for all y ∈ S 1 (x) implies that we have perfect matchings σ i : S 1 (x) → S 1 (y i ) for all 1 ≤ i ≤ d.In particular, we can assume that these perfect matchings σ i satisfy the following property: Property (P): If there exists a perfect matching between S 1 (x)\{y i } and S 1 (y i )\{x} then σ i (y i ) = x.
Our goal is to show that we can modify these perfect matchings in such a way that σ i (y j ) = σ j (y i ) for all i = j.Defining then η i : B 1 (x) → B 1 (y i ) as η i (x) = y i and η i (y) = σ i (y) for y ∈ S 1 (x) provide (S)-Ricci flatness.
We first prove the following crucial fact: Fact: Let i = j.We have σ i (y j ) = x if and only if y i and y j are not linked.
This fact can be shown as follows: We first prove the easier "⇐=" implication.Assume y i and y j are not linked.Then σ i (y j ) ∼ y i , y j cannot be in S 2 (x) and we must have therefore σ i (y j ) = x.For the "=⇒" implication, we provide an indirect proof: If y i and y j were linked by z ∈ S 2 (x), then the σ i -preimage of z ∈ S 1 (y i ) must be in {y i , y j } but we know that σ i (y j ) = x.Therefore σ i (y i ) = y j .Defining then the map σi : induces a perfect matching between S 1 (x)\{y i } and S 1 (y i )\{x}.This would imply σ i (y i ) = x contradicting to σ i (y j ) = x.Now we prove our goal.We first show that σ i (y j ) = x implies σ j (y i ) = x: Since σ i (y j ) = x, y i and y j are not linked by our Fact which, in turn, implies σ j (y i ) = x by our Fact, again.
We deal with all other pairs (i, j), i = j as follows: If σ i (y j ) = σ j (y i ), we do not change the assignments σ i (y i ), σ i (y j ), σ j (y i ), σ j (y j ).Now we assume that σ i (y j ) =: z = σ j (y i ) := z .Note that z, z ∈ S 2 (x) and they both are links of y i and y j .Since z ∈ S 1 (y j ) and d − x (z) ≤ 2, we must have σ −1 j (z) ∈ {y i , y j }.Since σ j is injective and σ j (y i ) = z , we must have σ −1 j (z) = y j .So we must have (13) σ j (y j ) = z.
Similarly, we conclude that σ i (y i ) = z .Now we modify σ i as follows: σ i (y i ) = z and σ i (y j ) = z .This preserves property (P) of the perfect matching σ i and establishes σ i (y j ) = σ j (y i ) for this pair of indices (i, j).
Note that if (i, j) and (k, l) are two different pairs with σ i (y j ) = σ j (y i ) and σ k (y l ) = σ l (y k ) then {i, j}∩{k, l} = ∅ for, otherwise, if k = i, there is no perfect matching between S 1 (x) and S 1 (y i ) since the four links between y i , y j and y i , y l can only have three possible preimages under σ i .This guarantees that we can repeat this process for all such pairs (i, j) simultaneously and we will end up with the required symmetric arrangement.
Finally, it remains to prove the forward implication of (b).The assumption κ LLY (x, y) = 2 d for all y ∈ S 1 (x) implies κ 0 (x, y) = 0 by Theorem 1.1.The existence of perfect matchings between S 1 (x)\{y i } and S 1 (y i )\{x} for all 1 ≤ i ≤ d from Remark 4.3 further imply that our chosen maps σ i satisfy σ i (y i ) = x for all i.In this situation, we can disregard the above possibility of z = σ i (y j ) = σ j (y i ) = z with z, z ∈ S 2 (x), since this would imply (13), which contradicts to σ j (y j ) = x.Therefore, the maps σ i do not need to be modified and the induced maps η i : B 1 (x) → V satisfy both symmetry and reflexivity.x (z) = 2 for all z ∈ S 2 (x) and κ 0 (x, y) < 0 for all y ∈ S 1 (x) as a counterexample.Note that S 1 (x) = {v 1 , . . ., v 6 }.
(b) All conditions in Theorem 4.5(a) are necessary: (i) If x is contained in a triangle, we have the icosidodecahedral graph (see Figure 1b) as a counterexample with κ 0 (x, y) = 0 for all edges {x, y} but K ∞ (x) < 0 for all vertices x, which means that x cannot be Ricci flat by Theorem 3.5.(ii) If we drop d − x (z) ≤ 2 for all z ∈ S 2 (x), Figure 5 provides a counterexample with κ 0 (x, y) = 0 for all y ∈ S 1 (x) and K ∞ (x) < 0.
(c) All conditions in Theorem 4.5(b) are necessary.Since in the case of triangles we have the following upper bound a natural generalization of the equivalence in the case of triangles would be the following statement: for all y ∈ S 1 (x) is equivalent to x being (RS)-Ricci flat.
(i) If x is contained in a triangle, we have K 3 × K 3 with d = 4 as a counterexample: x (z) ≤ 2 for all z ∈ S 2 (x), the 6-regular incidence graph of the (11,6,3)-design provides a counterexample with κ LLY (x, y) = 1  3 for all y ∈ S 1 (x), but x is not (RS)-Ricci flat (see Example 6.3).

Graph products
This section is concerned with three natural products of two graphs G and H: the tensor product G⊗H, the Cartesian product G×H, and the strong product G H. We will see that Ricci flatness is preserved under all three products.However, while Cartesian products preserve non-negativity of both Bakry-Émery and Ollivier Ricci curvature, we will see that this property fails to be true in the case of strong products.
Let us start with the definitions of these graph products: The vertex set of each of the three products G ⊗ H ( tensor product), G × H ( Cartesian product) and G H ( strong product) is given by V G × V H .To define the edge sets for each of these products, let denote the set of horizontal, vertical and diagonal edges.Then Our first result is concerned with preservance of Ricci flatness: Proof.Assume that G and H are Ricci flat at x ∈ V G and at y ∈ V H , respectively, that is, there exist maps satisfying the conditions (i),(ii),(iii) in Definition 3.1.
Note that we have the inclusions We define the following maps Note that We only consider the strong product case here, since all other products can be dealt with similarly by restrictions of the relevant η-maps to the corresponding 1-balls.We now check properties (i), (ii) and (iii) of Definition 3.1 for these maps on B G H 1 (x, y).
To verify (i), we observe that (u, v) Next, we verify (ii): The above observation implies that η i (u, v), η k (u, v) and η ⊗ jl (u, v) are mutually distinct for any choices of i, j, k, l.Moreover, it is easy to check that for any choice of i = j and k = l.Now we verify (iii): We have Similar commutation properties holds for the other families of η-maps, that is, we have * where η * and η * * are maps within the families η i , η k and η ⊗ jl .Combining these results, we obtain and In conclusion, Ricci flatness is preserved for all three graph products.
In the case of Cartesian products of two regular graphs G, H, there are explicit curvature formulas in terms of curvatures of the factors: Corollary 7.13] and Ollivier Ricci curvature κ G×H 0 (x, y) and κ G×H LLY (x, y) can be found in [14, Claim 1 and 2 in Proof of Theorem 3.1].In particular, non-negativity of each of these curvature notions is preserved under Cartesian products.In our next result, we provide lower curvature bounds for horizontal and vertical edges of the strong product G H: Theorem 5.3.Let G and H be two regular graphs with vertex degrees d G and d H , respectively.Lower Ollivier Ricci curvature bounds on horizontal edges and vertical edges are given by where κ * may refer to κ 0 or κ LLY and Proof of Theorem 5.3.Let us consider a horizontal edge (x 1 , y 1 ) ∼ (x 2 , y 1 ) where x 1 G ∼ x 2 .We will prove this argument for Lin-Lu-Yau curvature first.Let π G ∈ Π(µ ) be an optimal transport plan, i.e., its cost is equal to W G 1 (µ ).Now we define a function otherwise.
Now we verify the following marginal constraints showing that π is indeed a transport plan π ∈ Π(µ , µ (w 1 , z 1 ), and, similarly, The cost of this transport plan can then be calculated as Recall that π G is assumed to be an optimal transport plan and, therefore, , µ ).
Therefore, we have which gives the desired lower bound for κ 0 .
In the same way we obtain analogous results for vertical edges: Corollary 5.4.Let G and H be two regular graphs with non-negative κ 0 (or κ LLY ).Then all horizontal and vertical edges of G H have also non-negative κ 0 (or κ LLY ).
It turns out, however, that the statement of Corollary 5.4 is no longer true for diagonal edges, as the following example shows.
Another interesting question about graphs products is the following: In the case of Cartesian products, the full curvature function (as function of the dimension N ) at a vertex (x, y) is completely determined by the curvature functions of the factors at the vertices x and y (see [9, Theorem 7.9]): where * is a special operation defined in [9, Definition 7.1].We would like to know whether a similar formula (with a suitably defined operation) can be proved for tensor products and strong products.We can now apply Hall's Marriage Theorem to conclude that there is a perfect matching between S 1 (x)\{y} and S 1 (y)\{x}.
For the calculation of the Bakry-Émery curvature we employ the method presented at the beginning of Section 8 of [9] and the notation introduced there.In view of Theorem 8.1(i) in [9], we only need to verify that λ where ∆ S 1 (x) is the weighted Laplacian on the 1-sphere S 1 (x) with the following weights: for all y 1 , y 2 ∈ S 1 (x), y 1 = y 2 .
Since G is distance-regular, we obtain d − x (z) = c 2 and |{z ∈ S 2 (x) : This implies w y 1 y 2 = c 2 −1 c 2 and, therefore, the Laplacian ∆ It is tempting to assume that distance-regular graphs of girth 4 are always (R)-Ricci flat and then using Theorems 3.4 and 3.5(b) to conclude the statement of Theorem 6.2.However, the following example shows that this assumption is not always true.It remains an open question, however, whether every distance-regular graph of girth 4 is Ricci flat.Example 6.3 (Incidence graph of (11, 6, 3)-design).This is a distanceregular graph with intersection array {6, 5, 3; 1, 3, 6} (see [10]).
The structure of the incomplete 2-ball around a vertex x is given by: We give an indirect prove that this graph is not (R)-Ricci flat.Assume otherwise, i.e., there exists an associated matrix A with only 0 entries on diagonal.The other possible entries of A listed as below:

,
Recall that the matrix A cannot have repeated entries in any row and column.If the entry of A 12 is chosen to be 11, then all entries for the first three rows are uniquely determined as the numbers in red.Then the entry of A 46 cannot be either 7 or 14, due to appearance of them in the sixth column.Contradiction!Similarly, if the entry of A 12 is chosen to be 13, then all entries for the first three rows must be the numbers in blue.Then the entry of A 45 cannot be either 8 or 16 due to the fifth column.Contradiction!
However, the vertices of this graph are Ricci flat via the following matrix choice for A: We will show the following facts: (1) K d,d is (R)-Ricci flat for all d, (2) K d,d is (S)-Ricci flat for all d, (3) K d,d is (RS)-Ricci flat if and only if d is even.As before, we translate Ricci flatness properties at a vertex x, given by the maps η i , into properties of the associated d × d-matrix A = (A ij ).Since K d,d is triangle-free, we use a slightly different enumeration system for the matrix A: Let S 1 (x) = {y 1 , . . ., y d } where y j := η j (x), and S 2 (x) =: {z 1 , . . ., z t } and, furthermore, z 0 := x.Then the entries A ij ∈ {0, 1, . . ., t} of A are given via the relation and we have the following correspondences: (a) η i is injective corresponds to A ij = A ik for all j = k, (b) η i (y k ) = η j (y k ) corresponds to A ik = A jk for all i = j, (c) η 2 i (x) = x corresponds to A ii = 0, (d) η j (η i x) = η i (η j x) corresponds to A ji = A ij .In other words, (a) corresponds to the property that A has no repeated entries in the i-th row and (b) correspond to the property that A has no repeated entries in the k-th column.Moreover, (R)-Ricci flatness requires in addition that the matrix A has only the entry 0 on the diagonal, (S)-Ricci flatness requires that A is symmetric, and (RS)-Ricci flatness requires both additional properties of the matrix A. Note the general fact: (e) The number of occurrences of the entry m ∈ {0, . . ., t} in the matrix A is equal to d − x (z m ).( 1)-( 3) can now be shown by providing suitable matrices A. Proof of (1): , constructed as follows: • A ii = 0 for all 1 ≤ i ≤ 2n, • A ij = i + j − 2 for i = j and i + j ≤ 2n + 1, • A ij = i + j − 2n − 1 for i = j, i + j ≥ 2n + 2 and i, j ≤ 2n, • A i,2n = A 2n,i = 2(i − 1) for 1 ≤ i ≤ n, • A i,2n = A 2n,i = 2(i − n) − 1 for n + 1 ≤ i ≤ 2n − 1.Finally, assume that d is odd and x is (RS)-Ricci flat with associated symmetric matrix A with vanishing diagonal.Since d − x (z m ) = d for all m ∈ {0, . . ., t}, each entry m appears exactly d times in the matrix A by (e) above.Since d is odd and A symmetric, every entry must appear at least once on the diagonal, contradicting to the assumption of a vanishing diagonal.

(a)
The triplex (b) The icosidodecahedral graph

Figure 1 .
Figure 1.Examples of graphs with κ LLY = 0 We have the following possibilities for the entries of the associated matrix A:Note that (R)-Ricci flatness requires existence of an associated matrix A with vanishing diagonal and (S)-Ricci flatness requires existence of a symmetric matrix A. Therefore, x is (R)-and (S)-Ricci flat by the following matrix choices: Note that x is not (RS)-Ricci flat since both properties (vanishing diagonal and symmetry) cannot be satisfied at the same time.In fact, the complete bipartite graphs K d,d are both (R)and (S)-Ricci flat for all d, and (RS)-Ricci flat if and only if d is even (see the Appendix).

Figure 3 .
Figure 3.The incomplete 2-ball B inc 2 (x) of the Shrikhande graph

Proof of Theorem 4 . 5 .
The implications ⇐= in (a) and (b) follow immediately from Theorem 3.4 and Proposition 4.1.

Remark 4. 7 .
(a)  The reverse of the implication in Theorem 1.2(a) is not true since we have a triangle-free 2-ball in Figure4with K ∞ (x) = 0, d −

Theorem 5 . 2 .
Let G, H be two Ricci flat graphs.Then the graph products G ⊗ H, G × H and G H are again Ricci flat.Similarly, all three graph products preserve also (R)-Ricci flatness, (S)-Ricci flatness and (RS)-Ricci flatness.

Figure 7 .
Figure 7. Local Ollivier Ricci curvatures κ LLY of G and G P ∞ at edges incident to v 0 and (v 0 , w 0 ), respectively.Positive/negative/zero curvatures of edges is represented by the colours red/blue/grey.Every horizontal line of the lower graph represents a projection of G.

Figure 8 .
Figure 8. Local Bakry-Émery curvatures of G and G P ∞ at v 0 and (v 0 , w 0 ).Positive/negative curvatures of vertices is represented by the colours red/blue.Every horizontal line of the lower graph represents a projection of G.
where d H w is the vertex degree of w in H. Using distance-regularity, we obtain d H w = d H z = c 2 −1 and (17) implies |X| ≤ |Y |.
complete bipartite graphs K d,d

..
− 1 0 • • • d − 3 .Note that the first row of A is fixed and the following rows are obtained by a right shift of the previous row.Proof of (2):Note that the first row of A is fixed and the following rows are obtained by a left shift of the previous row.Proof of (3): Assume d = 2n even.Then we can choose A to be and again, κ 0 (x, y) ≥ 0, with equality if and only if N 1 = d − 1, which means T xy = ∅.