Edge-connectivity and non-negative Lin-Lu-Yau curvature

TL;DR

This paper proves that finite connected graphs with non-negative Lin-Lu-Yau curvature have edge connectivity equal to their minimum degree, answering an open question and classifying all such graphs with smaller edge connectivity.

Contribution

It establishes a key equality between edge connectivity and minimum degree for finite graphs with non-negative Lin-Lu-Yau curvature and classifies all exceptions.

Findings

Finite graphs with non-negative Lin-Lu-Yau curvature have edge connectivity equal to their minimum degree.

All connected graphs with non-negative curvature and smaller edge connectivity are infinite.

The result does not hold for infinite graphs, which are classified separately.

Abstract

By definition, the edge-connectivity of a connected graph is no larger than its minimum degree. In this paper, we prove that the edge connectivity of a finite connected graph with non-negative Lin-Lu-Yau curvature is equal to its minimum degree. This answers an open question of Chen, Liu and You. Notice that our conclusion would be false if we did not require the graph to be finite. We actually classify all connected graphs with non-negative Lin-Lu-Yau curvature and edge-connectivity smaller than their minimum degree. In particular, they are all infinite.

Equations

$$k^{\prime }(G)\le \delta (G),$$

$$k^{\prime }(G)\ge \delta (G)-1.$$

$$G\in \{\begin{array}{ll}{\mathcal{G}}_{k^{\prime }(G)},& \text{if }{k}^{\prime }(G)=1,2\text{ or }{k}^{\prime }(G)\ge 5,\\ \{G_3^{\ast }\}\cup {\mathcal{G}}_3,& \text{if }{k}^{\prime }(G)=3,\\ \{G_4^{\ast }\}\cup {\tilde{\mathcal{G}}}_4,& \text{if }{k}^{\prime }(G)=4.\end{array}$$

$$S_1(e{)}_{(G)}:=\{ϵ\in E(G)|\text{e and ϵ share a common vertex}\}.$$

$$\underset{e\in E(H)}{\min }|S_1(e{)}_{(H)}|\le |E(H)|-\frac{|V(H)|}{2}.$$

$$K_{2,2},K_4,K_4-e,K_3+e,n{K}_{1,1}\text{for }n\ge 2,2K_{1,\frac{n}{2}}\text{for even }n>2,S{T}_{\frac{n-1}{2},\frac{n-1}{2}}\text{for odd }n\ge 3.$$

$$d_x{d}_y{\kappa }_{LLY}(e)+d_x\left(2|E(H)|-|V(H)|-2|S_1(e{)}_{(H)}|\right)\le 2d_x\left(|E(H)|+1-d_y\right).$$

$$k^{\prime }(G)=|E(H)|\ge \delta (G)-1.$$

$$K_{2,2},n{K}_{1,1}\text{for }n\ge 2,2K_{1,\frac{n}{2}}\text{for even }n>2,S{T}_{\frac{n-1}{2},\frac{n-1}{2}}\text{for odd }n\ge 3\text{or}{K}_{1,n}\text{for }n\ge 1.$$

$$K_{2,2},n{K}_{1,1}\text{for }n\ge 2,2K_{1,2},S{T}_{1,1}\text{and}{K}_{1,n}\text{for }n\ge 1.$$

$$S_1(u)=\{w\in V:d(u,w)=1\},\text{and}{B}_1(u)=\{w\in V:d(u,w)\le 1\}.$$

$$W({\mu }_1,{\mu }_2)=\underset{\pi }{\inf }\sum _{u\in V(G)}\sum _{v\in V(G)}d(u,v)\pi (u,v),$$

$$\sum _{v\in V(G)}\pi (u,v)={\mu }_1(u)\text{and}\sum _{u\in V(G)}\pi (u,v)={\mu }_2(v),\text{for any }u,v\in V(G).$$

$$W({\mu }_1,{\mu }_2)=\underset{\begin{array}{c}f:V(G)\to \mathbb{R}\\ \text{1-Lipschitz}\end{array}}{\sup }\sum _{u\in V(G)}f(u)\left({\mu }_1(u)-{\mu }_2(u)\right),$$

$${\mu }_x^{\rho }(v)=\{\begin{array}{ll}\rho ,& \text{if }v=x,\\ \frac{1-\rho }{d_x},& \text{if }v\sim x,\\ 0,& \text{otherwise.}\end{array}$$

$${\kappa }_{\rho }(x,y)=1-\frac{W({\mu }_x^{\rho },{\mu }_y^{\rho })}{d(x,y)}.$$

$${\kappa }_{LLY}(x,y)=\underset{\rho \to 1}{\lim }\frac{{\kappa }_{\rho }(x,y)}{1-\rho }.$$

$$\begin{gathered} {\kappa }_{LLY}(e)=\frac{{\kappa }_{\rho }(x,y)}{1-\rho } \\ =\frac{1}{1-\rho }\left(1-W\left({\mu }_x^{\rho },{\mu }_y^{\rho }\right)\right),\text{for any}\rho \in \left[\frac{1}{\max \{d_x,d_y\}+1},1\right). \end{gathered}$$

$$\pi (u,\varphi (u))=\frac{1}{\beta +1}$$

$$(\beta +1)\cdot W({\mu }_x,{\mu }_y)=\underset{\psi :B_1(x)\setminus B_1(y)\to B_1(y)\setminus B_1(x)}{\min }\sum _{u\in B_1(x)\setminus B_1(y)}d(u,\psi (u)),$$

$$(d_u-1)+(d_v-1)=|S_1(e{)}_{(H)}|>|E(H)|-\frac{|V(H)|}{2}=\frac{1}{2}\sum _{w\in V(H)}(d_w-1).$$

$$\sum _{w\in V(H)}(d_w-1)\ge \sum _{i=1}^2\left[(d_{u_i}-1)+(d_{v_i}-1)\right]>\sum _{w\in V(H)}(d_w-1),$$

$$\sum _{w\in V(H)}(d_w-1)=\sum _{i=1}^2\left[(d_{u_i}-1)+(d_{v_i}-1)\right].$$

$$0=|S_1(e)|\ge |E(H)|-\frac{|V(H)|}{2}\ge 0.$$

$$L:=\{u\in V(H):d_u=1\}\text{and}S:=\{w\in V(H):\text{there exists }u\in L\text{such that}w\sim u\}.$$

$$d_w-1\ge |E(H)|-\frac{|V(H)|}{2},\text{for any }w\in S.$$

$$2|E(H)|=\sum _{w\in V(H)}{d}_w\ge |L|+|S|\left(|E(H)|-\frac{|V(H)|}{2}+1\right)+2|V\setminus (L\cup S)|.$$

$$|S|\le 2-\frac{2|V(H)\setminus (L\cup S)|}{2|E(H)|-|V(H)|}.$$

$$|V(H_i)|\ge d_{x_i}+d_{y_i}=1+d_{y_i}=|S(e_i)|+2\ge |E(H)|-\frac{|V(H)|}{2}+2=\frac{|E(H)|}{2}+1,\text{for }i=1,2.$$

$$|E(H)|+2=|V(H)|=|V(H_1)|+|V(H_2)|\ge |E(H)|+2.$$

Full text

1. Introduction and statement of results

The interplay between local and global properties of spaces has long been a central theme in the study of both geometry and graph theory. The local properties of spaces are often described by curvature bounds in geometry [14] and by local graphs in graph theory [10, 30, 31]. With various synthetic notions of discrete curvature, the approaches of the two disciplines have interacted quite deeply [21]. In this paper, we explore connections between local properties as captured by the Lin–Lu–Yau curvature and global properties—edge-connectivity and finiteness—of locally finite graphs.

The edge-connectivity $k^{\prime}(G)$ of a locally finite graph $G$ with at least two vertices is the minimum cardinality of an edge cut of $G$. Here, an edge cut is a set of edges whose deletion increases the number of connected components. We call an edge cut with the minimum cardinality a min-cut of $G$. The edge-connectivity of a single vertex is defined to be 0. (Notice that the notation $k(G)$ is reserved for the vertex-connectivity of a graph $G$.) By definition, we have

[TABLE]

where $\delta(G)$ stands for the minimum vertex degree of $G$. This estimate is sharp. In fact, it has been shown that the equality holds for any finite connected edge-transitive graphs [19, 29], vertex-transitive graphs [19], and distance-regular graphs [4, 5]. By a recent result [6], the equality still holds for any (possibly infinite) connected amply regular graphs, for which any two vertices at distance $2$ have more than one common neighbor. On the other hand, the gap between the edge-connectivity and minimum vertex degree can be arbitrarily large. Indeed, for any integers $0<\ell\leq d$, there exists a graph $G$ with $k^{\prime}(G)=\ell$ and $\delta(G)=d$.

The Lin–Lu–Yau curvature is a discrete analogue of the Ricci curvature in Riemannian geometry. It was proposed by Lin, Lu and Yau [17] as a modification of Ollivier’s notion of coarse Ricci curvature [22]. The Lin–Lu–Yau curvature of an edge $x\sim y\in E$ is defined via comparing the two neighborhoods around $x$ and $y$ in terms of Wasserstein distance. Intuitively, the curvature along an edge $x\sim y\in E$ is positive (resp., non-negative) when the Wasserstein distance between the two neighborhoods is strictly less than (resp., no larger than) the combinatorial distance between $x$ and $y$. A locally finite graph $G$ is said to have positive (resp., non-negative) Lin–Lu–Yau curvature if every edge in $G$ has positive (resp., non-negative) Lin–Lu–Yau curvature. Moreover, a graph has positive (resp. non-negative) Lin–Lu–Yau curvature precisely when it has positive (resp. non-negative) $\frac{1}{2}$-Ollivier Ricci curvature, see (2.3) below. There is a large body of literature on various properties of graphs under Ollivier or Lin–Lu–Yau curvature bounds, see, e.g., [1, 2, 3, 8, 12, 13, 15, 16, 18, 20, 23, 24, 25].

The interaction between vertex- and edge-connectivity and Lin–Lu–Yau curvature of connected graphs has been explored by Chen, the first named author and You [7]. In particular, they establish the following relationship between the Lin–Lu–Yau curvature and edge-connectivity. Notice that all graphs in this paper are locally finite and simple.

Theorem 1.1**.**

[7]** Let $G$ be a connected graph with minimum vertex degree $\delta(G)$ and edge-connectivity $k^{\prime}(G)$. If G has positive Lin–Lu–Yau curvature, then $k^{\prime}(G)=\delta(G)$.

Observe that one can not weaken the assumption of positive Lin–Lu–Yau curvature in Theorem 1.1 to be non-negative Lin–Lu–Yau curvature. For example, the infinite path graph has non-negative Lin–Lu–Yau curvature, yet its edge-connectivity equals $1$ despite having a constant vertex degree of $2$. Chen, the first named author and You [7] posed the question of whether any finite connected graph with non-negative Lin–Lu–Yau curvature satisfies $k^{\prime}(G)=\delta(G)$.

In this paper, we completely answer this problem.

Theorem 1.2**.**

Let $G$ be a finite connected graph with minimum vertex degree $\delta(G)$ and edge-connectivity $k^{\prime}(G)$. If $G$ has non-negative Lin–Lu–Yau curvature, then $k^{\prime}(G)=\delta(G)$.

We remark that graphs with positive or non-negative Lin–Lu–Yau curvature can be infinite. It is quite difficult to apply the global finiteness assumption to improve the original proof of Theorem 1.1 in [7]. To establish Theorem 1.2, we find an approach different from that used in [7] and first show the following estimate.

Theorem 1.3**.**

Let $G$ be a connected graph with minimum vertex degree $\delta(G)$ and edge-connectivity $k^{\prime}(G)$. If $G$ has non-negative Lin–Lu–Yau curvature, then

[TABLE]

Notice that, the above estimate tells that any connected graph with non-negative Lin–Lu–Yau curvature satisfies $k^{\prime}(G)\in\{\delta(G)-1,\delta(G)\}$. Furthermore, we characterize all graphs achieving the equality in Theorem 1.3.

Theorem 1.4**.**

Let $G$ be a connected graph with minimum vertex degree $\delta(G)$ and edge-connectivity $k^{\prime}(G)$. Then $G$ has non-negative Lin–Lu–Yau curvature with $k^{\prime}(G)=\delta(G)-1$ if and only if

[TABLE]

Theorem 1.4 is our main contribution. The sets of graphs, $\mathcal{G}_{n},n\in\mathbb{Z}_{>0}$ and $\tilde{\mathcal{G}}_{4}$, as well as the two specific graphs $G_{3}^{*}$ and $G_{4}^{*}$, will be defined in Subsection 1.1. Notably, $\mathcal{G}_{1}=\{P_{\infty}\}$, where $P_{\infty}$ denotes the infinite path graph, and moreover the Cartesian product $P_{\infty}\times K_{n}$, with $K_{n}$ the complete graph on $n$ vertices, belongs to the set $\mathcal{G}_{n}$. In fact, each graph listed in (1.2) is infinite. Therefore, Theorem 1.2 is a direct consequence of Theorem 1.4 together with Theorem 1.3.

Moreover, each graph listed in (1.2) has exactly two ends. Hua and Münch [12] proved that any connected graph with non-negative Lin–Lu–Yau curvature has at most two ends. Hence, our results indicate that any connected graph $G$ with non-negative Lin–Lu–Yau curvature and at most one end satisfies $k^{\prime}(G)=\delta(G)$. On the other hand, there are connected graphs with non-negative Lin–Lu–Yau curvature and exactly two ends for which $k^{\prime}(G)=\delta(G)$.

We will outline the approach used to prove Theorem 1.3 and Theorem 1.4 in Subsection 1.2.

Before diving into more details, we would like to remark on other related works. Horn, Purcilly and Stevens [11] established a lower bound estimate for the vertex-connectivity in terms of the minimum vertex degree and the Bakry–Émery curvature lower bounds of the graph. Bakry–Émery curvature is another discrete notion of Ricci curvature. Notice that Bakry–Émery curvature and Lin–Lu–Yau curvature can behave very differently. There are graphs whose Bakry–Émery curvature and Lin–Lu–Yau curvature have opposite signs. Chen, Koolen and the first named author [6] showed that every connected graph with non-negative Bakry–Émery curvature satisfies $k^{\prime}(G)\geq\delta(G)-1$. Consequently, any connected regular graph with non-negative Bakry–Émery curvature and an even or infinite number of vertices has a perfect matching, see [6, Theorem 1.1]. Indeed, this perfect matching statement is a direct consequence of the edge-connectivity estimate $k^{\prime}(G)\geq\delta(G)-1$ combined with Tutte’s theorem [26, 27]. Theorem 1.3 is a counterpart of their result in terms of Lin–Lu–Yau curvature. Using the same argument, we obtain the following corollary.

Corollary 1.5**.**

Let $G$ be a connected regular graph with an even or infinite number of vertices. If $G$ has non-negative Lin–Lu–Yau curvature, then $G$ has a perfect matching.

1.1. Graphs listed in Theorem 1.4

We first prepare necessary notations. Let $k,\ell$ be two positive integers. Recall that $K_{k}$ denotes the complete graph on $k$ vertices. Let $K_{1,k}$ denote the star graph with $k$ leaves. A double star graph $ST_{k,\ell}$ is a tree with a central edge $e=x\sim y$, where $x$ has $k$ leaves attached and $y$ has $\ell$ leaves attached. For a graph $G$, the notation $kG$ represents $k$ disjoint copies of $G$.

We write $V(G)$ and $E(G)$ for the vertex set and edge set of $G$, respectively. For any subsets $S,T\subset V(G)$, we denote by $G[S]$ the induced subgraph on $S$ in $G$; Let $E(S,T):=\{u\sim v:u\in S\ \text{and}\ v\in T\}$ be the set of edges between $S$ and $T$; We denote by $G[S,T]$ the subgraph of $G$ with $V(G[S,T])=S\cup T$ and $E(G[S,T])=E(S,T)$. Notably, we have $G[S]=G[S,S]$. Notice that the subgraph $G[S,T]$ does not always coincide with the edge-induced subgraph of $E(S,T)$. For example, in Figure 1, the edge-induced subgraph of $E(S,T)$ is the complete bipartite graph $K_{2,2}$, whereas the subgraph $G[S,T]$ consists of $K_{2,2}$ and two additional isolated vertices.

Definition 1.6** (The graph set $\mathcal{G}_{n}$).**

For any positive integer $n$, let $\mathcal{G}_{n}$ be the set of graphs $G$ satisfying the following properties: There exists a partition of the vertex set $V(G)=\sqcup_{i\in\mathbb{Z}}V_{i}$, such that

• (i)

$G[V_{i}]\in\{K_{1},K_{n}\}$ for any $i\in\mathbb{Z}$;

• (ii)

$E(V_{i},V_{j})=\emptyset$ for any $|i-j|>1$;

• (iii)

$G[V_{i},V_{i+1}]\in\{K_{1,n},nK_{1,1}\}$ for any $i\in\mathbb{Z}$.

Note that $\mathcal{G}_{1}=\{P_{\infty}\}$ and $P_{\infty}\times K_{n}\in\mathcal{G}_{n}$ for any positive integer $n$. Next, we define the two graphs $G_{3}^{*}$ and $G_{4}^{*}$.

Definition 1.7** (The graphs $G_{3}^{*}$).**

We define $G_{3}^{*}$ to be the $4$-regular graph satisfying the following properties: There exists a partition of the vertex set $V(G)=\sqcup_{i\in\mathbb{Z}}V_{i}$, such that

• (i)

$G[V_{i}]=K_{2}$ for any $i\in\mathbb{Z}$;

• (ii)

$E(V_{i},V_{j})=\emptyset$ for any $|i-j|>1$;

• (iii)

$G[V_{i},V_{i+1}]=ST_{1,1}$ for any $i\in\mathbb{Z}$.

Definition 1.8** (The graphs $G_{4}^{*}$).**

We define $G_{4}^{*}$ to be the $5$-regular graph satisfying the following properties: There exists a partition of the vertex set $V(G)=\sqcup_{i\in\mathbb{Z}}V_{i}$, such that

• (i)

$G[V_{i}]=K_{4}$ for any $i\in\mathbb{Z}$;

• (ii)

$E(V_{i},V_{j})=\emptyset$ for any $|i-j|>1$;

• (iii)

The edge-induced subgraph of $E(V_{i},V_{i+1})$ is isomorphic to $2K_{1,2}$ with $3$ vertices in $V_{i}$ and the other $3$ vertices in $V_{i+1}$.

The two graphs $G_{3}^{*}$ and $G_{4}^{*}$ are depicted in Figure 3 and Figure 3.

The graph set $\tilde{\mathcal{G}}_{4}$ is defined as follows.

Definition 1.9** (The graph set $\tilde{\mathcal{G}}_{4}$).**

We define $\tilde{\mathcal{G}}_{4}$ as the set of graphs $G$ satisfying the following properties: There exists a partition of the vertex set $V(G)=\sqcup_{i\in\mathbb{Z}}V_{i}$, such that

• (i)

$G[V_{i}]\in\{K_{1},K_{2},K_{4}\}$ for any $i\in\mathbb{Z}$;

• (ii)

$E(V_{i},V_{j})=\emptyset$ for any $|i-j|>1$;

• (iii)

$G[V_{i},V_{i+1}]\in\{K_{1,4},4K_{1,1},K_{2,2},2K_{1,2}\}$ for any $i\in\mathbb{Z}$.

• (iv)

$(|V_{i-1}|,|V_{i}|,|V_{i+1}|)\neq(4,2,4)$ for any $i\in\mathbb{Z}$.

Remark 1.10**.**

Condition (iv) in Definition 1.9 excludes the existence of edges with negative Lin–Lu–Yau curvature in the graph. Let $G$ be a graph whose vertex set can be partitioned as $V(G)=\sqcup_{i\in\mathbb{Z}}V_{i}$ satisfying conditions (i)–(iii). Suppose that for some index $i$ we have $(|V_{i-1}|,|V_{i}|,|V_{i+1}|)=(4,2,4)$. Condition (iii) implies that both $G[V_{i-1},V_{i}]$ and $G[V_{i},V_{i+1}]$ are isomorphic to $2K_{1,2}$, and the resulting local structure is depicted in Figure 4. Notice that each edge in $E(V_{i-1},V_{i})$ or $E(V_{i},V_{i+1})$ has Lin–Lu–Yau curvature equal to $-0.2$.

Observe that $\mathcal{G}_{4}$ is a subset of $\tilde{\mathcal{G}}_{4}$. To illustrate the difference between these two sets, we depict in Figure 6 a graph that belongs to $\mathcal{G}_{4}$ but is not the Cartesian product $P_{\infty}\times K_{n}$, whereas in Figure 6 we depict a graph in $\tilde{\mathcal{G}}_{4}\setminus\mathcal{G}_{4}$.

1.2. Strategy of proving the main results

We begin by presenting a combinatorial inequality for graphs without isolated vertices and examining its rigidity properties. This inequality serves as a crucial tool for proving the edge-connectivity bound in Theorem 1.3 and for addressing the rigidity problem discussed in Theorem 1.4. For an edge $e\in E(G)$, define

[TABLE]

The inequality and its rigidity characterization can be described as follows.

Theorem 1.11**.**

Let $H$ be a finite graph with no isolated vertices, and suppose $H$ is neither a star graph nor $K_{3}$. Then

[TABLE]

Moreover, equality in (1.3) holds if and only if $H$ is one of the following:

[TABLE]

Here, $K_{4}-e$ denotes the graph obtained by removing one edge from $K_{4}$, while $K_{3}+e$ is formed by adding a new vertex and connecting it to one of the vertices of $K_{3}$.

Another crucial ingredient is a Lin–Lu–Yau curvature estimate for an edge in a cut. Let $G$ be a connected graph and let $E_{0}$ be a min-cut. Let $H$ be the edge-induced subgraph of $G$ induced by the edges in $E_{0}$. This subgraph $H$ is bipartite and contains no isolated vertices. In Theorem 4.1, we will show that for any edge $e=x\sim y\in E_{0}=E(H)$ with $d_{x}\geq d_{y}$, the Lin–Lu–Yau curvature $\kappa_{LLY}(e)$ satisfies

[TABLE]

By combining this inequality with Theorem 1.11, we deduce that if $G$ has non-negative Lin–Lu–Yau curvature and $H$ is not a star graph, then

[TABLE]

Note that $H$ cannot be $K_{3}$ since it is bipartite. In the case where $H$ is a star graph, a sharper curvature bound is required (see Theorem 4.2). All together lead to a proof of Theorem 1.3.

Now, suppose $G$ is connected with non-negative Lin–Lu–Yau curvature and $k^{\prime}(G)=\delta(G)-1$. Then equality must hold in both (1.3) and (1.4). The rigidity statement in Theorem 1.11 implies that the edge-induced subgraph $H$ of a min-cut in $G$ can only be one of the following:

[TABLE]

The rigidity result corresponding to (1.4) (see Theorem 4.1) provides further insight into the local configuration around $H$. Moreover, the above set of possible graphs for $H$ can be further reduced to be (see Lemma 5.2)

[TABLE]

The remaining question is whether these local patterns can be extended to the entire graph. We will explore this issue in detail in Section 5. The interplay between curvature bounds for edges in a cut and the combinatorial inequality (1.3) is particularly noteworthy.

The rest of this paper is organized as follows: In Section 2, we introduce the notations and review basics about Lin–Lu–Yau curvature of graphs. Section 3 contains the proof of Theorem 1.11. In Section 4, we establish curvature estimates of edges in a cut, see Theorem 4.1 and Theorem 4.2. In particular, we prove (1.4). Finally, in Section 5, we complete the proofs of Theorem 1.3 and Theorem 1.4. In Appendix A, we check that each graph listed in Theorem 1.4 has non-negative Lin–Lu–Yau curvature.

2. Preliminaries

Let $G$ be a graph with vertex set $V(G)$ and edge set $E(G)$. For any $u,v\in V(G)$, we denote by $d(u,v)$ the combinatorial distance between $u$ and $v$, and define

[TABLE]

The cardinality of $S_{1}(u)$ is the vertex degree $d_{u}$ of $u$. For any $u\in V(G)$ and $X\subset V(G)$, we define $d(u,X):=\min_{v\in X}d(u,v)$. Throughout this paper, all graphs are simple and locally finite, that is, there are no self-loops or multi-edges and the degree of each vertex is finite.

Next, we recall basics about Ollivier Ricci curvature and Lin–Lu–Yau curvature of graphs. The definition is based on the concept of the Wasserstein distance between probability measures.

Definition 2.1** (Wasserstein Distance).**

Let $G$ be a graph and $\mu_{1},\mu_{2}$ be two probability measures on $V(G)$. The Wasserstein distance $W(\mu_{1},\mu_{2})$ is defined as

[TABLE]

where the infimum is taken over all transport plans $\pi:V(G)\times V(G)\to[0,1]$ satisfying the marginal constraints:

[TABLE]

The Kantorovich duality tells that (see, e.g.,  [28, Theorem 1.14])

[TABLE]

where the supremum is taken over all $1$-Lipschitz functions $f$. Recall that a function $f:V(G)\to\mathbb{R}$ is called $1$-Lipschitz if $|f(x)-f(y)|\leq d(x,y)$ holds for any $u,v\in V(G).$

For a vertex $x$ and an idleness parameter $\rho\in[0,1]$, we define the probability measure $\mu_{x}^{\rho}$ as

[TABLE]

Definition 2.2** (Ollivier Ricci curvature [22, 23] and Lin–Lu–Yau curvature [17]).**

Let $G$ be a graph. For $x,y\in V(G)$, the $\rho$-Ollivier Ricci curvature is

[TABLE]

The Lin–Lu–Yau curvature $\kappa_{LLY}(x,y)$ is then defined by the limit

[TABLE]

Notice that $\kappa_{LLY}(x,y)=-\left.\frac{d\kappa_{\rho}(x,y)}{d\rho}\right|_{\rho=1}$ since $\kappa_{1}(x,y)=0$. For an edge $e=x\sim y$, it has been shown in [3] that the function $\rho\mapsto\kappa_{\rho}(x,y)$ is linear on the interval $\left[\frac{1}{\max\{d_{x},d_{y}\}+1},1\right]$. This yields a limit-free expression for $\kappa_{LLY}(e)$:

[TABLE]

In case that $d_{x}=d_{y}=:\beta$, we have $\mu_{x}(u),\mu_{y}(u)\in\{0,\frac{1}{\beta+1}\}$ for any $u\in V(G)$. Then it is a Monge problem to calculate $W(\mu_{x},\mu_{y})$. That is, an optimal transport plan $\pi$ between $\mu_{x}$ and $\mu_{y}$ can be induced by a bijective map $\phi:B_{1}(x)\to B_{1}(y)$ via

[TABLE]

for any $u\in B_{1}(x)$, see [8, Proposition 2.7] and [20, Theorem 2.6]. By [3, Lemma 4.1], we can choose a map $\phi$ for an optimal transport plan such that $\phi(u)=u$ for any $u\in B_{1}(x)\cap B_{1}(y)$. That is, we have for an edge $e=x\sim y$ with $d_{x}=d_{y}=:\beta$ that

[TABLE]

where the minimum is taken over all bijective maps between $B_{1}(x)\setminus B_{1}(y)$ and $B_{1}(y)\setminus B_{1}(x)$.

Recall that by triangle inequality for Wasserstein distance, $\kappa_{LLY}(x,y)\geq 0$ for all $x,y\in V(G)$ if and only if $\kappa_{LLY}(e)\geq 0$ for all $e\in E(G)$, see [17, Lemma 2.3].

3. A combinatorial inequality and its rigidity

In this section, we prove Theorem 1.11.

Proof.

We first prove the inequality (1.3) by contradiction. Suppose that (1.3) does not hold. That is, we have for each edge $e=u\sim v\in E(H)$ that

[TABLE]

Since $H$ has no isolated vertices and is not a star graph, $H$ has at least two edges. If any two edges of $H$ have a common vertex, i.e., the line graph of $H$ is a complete graph, then $H$ is either a star graph or $K_{3}$. Therefore, by our assumption, there exist two disjoint edges $u_{i}\sim v_{i}$, $i=1,2$ in $E(H)$. Applying (3.1) leads to

[TABLE]

which is a contradiction. This proves (1.3).

Assume that the equality holds in (1.3), that is, $\min_{e\in E(H)}|S_{1}(e)_{(H)}|=|E(H)|-\frac{|V(H)|}{2}$. Then we derive by the above argument that

[TABLE]

Since $H$ has no isolated vertices and is not a star graph or $K_{3}$, we have $|V(H)|\geq 4$. If $|V(H)|=4$, there are only $6$ possible graphs with no isolated vertices which are not star graphs. It is direct to check that the equality in (1.3) holds for all of them: $2K_{1,1},K_{4},K_{4}-e,K_{2,2},K_{3}+e$ and $ST_{1,1}$. Note that $ST_{1,1}$ is a path graph with $4$ vertices.

Next, we assume $|V(H)|>4$. Due to (3.2), every vertex in $V(H)\setminus\{u_{1},u_{2},v_{1},v_{2}\}$ is of degree $1$. If there exists an edge $e=x\sim y$ such that $x,y\in V(H)\setminus\{u_{1},u_{2},v_{1},v_{2}\}$, then we derive from the equality in (1.3) that

[TABLE]

This forces that $2|E(H)|=\sum_{w\in V(H)}d_{w}=|V(H)|$. Therefore, every vertex in $H$ is of degree $1$, that is, $H=nK_{1,1}$ with $n\geq 3$.

If, otherwise, there does not exist an edge $e=x\sim y$ with $x,y\in V(H)\setminus\{u_{1},u_{2},v_{1},v_{2}\}$, then every vertex in $V(H)\setminus\{u_{1},u_{2},v_{1},v_{2}\}$ is connected to one of $\{u_{1},u_{2},v_{1},v_{2}\}$ via an edge. In particular, $H$ has at most $2$ connected components. Moreover, since at least one of $\{u_{1},u_{2},v_{1},v_{2}\}$ has degree at least $2$, we have $2|E(H)|>|V(H)|$. Denote by

[TABLE]

Note that $S\subset\{u_{1},u_{2},v_{1},v_{2}\}$ is not empty. Then we deduce from the equality in (1.3) that

[TABLE]

Therefore, we derive

[TABLE]

Rearranging leads to

[TABLE]

Notice that $|S|>1$. Indeed, if $|S|=1$, there exists $i\in\{1,2\}$ such that $d_{u_{i}}=d_{v_{i}}=1$. By an argument similar as in (3.3), we have $|V(H)|=2|E(H)|$, which is a contradiction. Therefore, we have $|S|=2$ and $V(H)\setminus{(L\cup S)}=\emptyset$. This tells that $H$ is a forest. In the following, we divide our discussion into two cases.

Case 1: $H$ has $2$ connected components $H_{1}$ and $H_{2}$. Then we have $|V(H)|-|E(H)|=2$. Let $x_{i}$ be a leaf of $H_{i}$ and $y_{i}\in V(H_{i})$ be a neighbor of $x_{i}$. Denote the edge by $e_{i}=x_{i}\sim y_{i}$. Then

[TABLE]

Summing up leads to

[TABLE]

This forces that $V(H_{i})=1+d_{y_{i}}=1+\frac{|E(H)|}{2}$. Hence, $|E(H)|$ is even and $H_{i}$ is a star graph with $\frac{|E(H)|}{2}+1$ vertices. That is, we have $H=2K_{1,\frac{n}{2}}$ with $n=|E(H)|>2$ even.

Case 2: $H$ is connected, that is, $H$ is a tree. In this case, we have $|V(H)|-|E(H)|=1$. For convenience, we denote by $S:=\{y_{1},y_{2}\}\subset\{u_{1},u_{2},v_{1},v_{2}\}$. Then, we must have $y_{1}\sim y_{2}$. Since both are connected to a vertex of degree $1$, we estimate

[TABLE]

Summing up, we deduce

[TABLE]

This forces that $d_{y_{i}}=\frac{|V(H)|}{2}$, $i=1,2$. This tells that $H=ST_{\frac{n-1}{2},\frac{n-1}{2}}$ with $n=|V(H)|-1=|E(H)|$. Since $d_{y_{i}}$ is an integer, $n=|E(H)|=|V(H)|-1>3$ is odd.

In conclusion, if the equality in (1.3) holds, then $H$ is one of the following graphs: $K_{2,2}$, $K_{4}$, $K_{4}-e$, $K_{3}+e$; $nK_{1,1}$ for $n\geq 2$, $2K_{1,\frac{n}{2}}$ for $n>2$ even, $ST_{\frac{n-1}{2},\frac{n-1}{2}}$ for $n\geq 3$ odd. It is straightforward to check the equality in (1.3) does hold for all those graphs. This completes the proof. ∎

4. Lin–Lu–Yau curvature of edges in a cut

In this section, we prove the following theorem relating (1.3) to the Lin–Lu–Yau curvature.

Theorem 4.1**.**

Let $G=(V,E)$ be a connected graph. Let $E_{0}\subset E$ be a cut of $G$ and $H$ be the edge-induced subgraph on $E_{0}$ of $G$. For any edge $e=x\sim y\in E_{0}=E(H)$, we have

[TABLE]

If equality is achieved, then every vertex $u\in A\setminus\{x\}$ (resp., $v\in B\setminus\{y\}$) is adjacent to $x$ (resp., $y$). Here, $A$ (resp., $B$) denotes the set of vertices in $V(H)$ that lie in the same connected component as $x$ (resp., $y$) once the cut $E_{0}$ is removed.

Proof.

Recall that $S_{1}(e)_{(H)}$ is the set of edges in $E_{0}=E(H)$ incident to the edge $e=x\sim y$. We denote by

[TABLE]

Note that $|M|=|S_{1}(e)_{(H)}|$.

Without loss of generality, we assume that $d_{x}\geq d_{y}$. Denote by $a:=\frac{1}{d_{x}(d_{y}+1)}$. Consider the following two probability measures as defined in (2.2):

[TABLE]

Let $\pi$ be a transport plan such that

[TABLE]

By [3, Lemma 4.1], we can further assume that

[TABLE]

This implies that $\pi(x,v)=\pi(v,x)=0$ for any $v\neq x$, and $\pi(y,v)=0$ for any $v\neq y$. Let us define the following sets:

[TABLE]

By definition, we have

[TABLE]

Observing from (4.2) that $\sum_{u\not\in M}\pi(u,v)\leq(d_{x}-d_{y})a$ for any $v\in B_{1}(x)\cap B_{1}(y)\cap M$. This leads to the following estimate:

[TABLE]

Inserting (4) into (4.3) and rearranging yield that

[TABLE]

We further consider the following two subsets of $Z_{3}$,

[TABLE]

Recall that $E_{0}=E(H)$ is a cut of $G$. Let $V(G)=X\sqcup Y$ be the partition of $V(G)$ such that $x\in X$, $y\in Y$ and $E(X,Y)=E_{0}=E(H)$. This leads to the following observations: For any $(u,v)\in Z_{3,1}$, we have $u\in X$, $v\in Y$ and hence the edge $u\sim v$ must lie in $E_{0}=E(H)$. That is, both $u$ and $v$ lie in $V(H)$; For any $u\in Z_{3,2}$, we have $u\in X$, $v\in Y$ and hence the shortest $2$-path connecting $u$ and $v$ in $G$ contains exactly one edge in $E_{0}=E(H)$. That is, at least one of $u$ and $v$ lies in $V(H)$. Then, we derive that

[TABLE]

Now we estimate the optimal transportation distance

[TABLE]

Applying (4.5) and (4) leads to

[TABLE]

By (2.3), we obtain

[TABLE]

This proves (4.1). Moreover, if the equality in (4.1) holds, then the equalities in (4), (4) and (4.7) must hold. Since the equality (4.7) holds, we have $\pi(u,v)d(u,v)=0$ for any $(u,v)\in Z_{1}\sqcup Z_{2}$. This tells in particular that $\pi(u,v)=0$ for any $(u,v)\in Z_{2}$, and $\pi(u,v)\neq 0$ only if $d(u,v)=0$. Then, we applying the equality in (4) to deduce

[TABLE]

Therefore, we have $d_{x}=d_{y}=\pi(u,u)$ for any $u\in M$ if $M\neq\emptyset$. This implies that $M\subset B_{1}(x)\cap B_{1}(y)$.

We further apply the equality in (4) to conclude that $u\sim x$ for any $u\in A\setminus\{x\}$ and $v\sim y$ for any $v\in B\setminus\{y\}$, where $A:=X\cap V(H)$ and $B:=Y\cap V(H)$. This completes the proof of this theorem. ∎

Theorem 4.2**.**

Let $G=(V,E)$ be a connected graph. Let $E_{0}\subset E$ be a cut of $G$ and $H$ be the edge-induced subgraph of $E_{0}$ in $G$. Suppose that $H$ is a star graph. Then we have for any edge $e=x\sim y\in E_{0}=E(H)$ with $y$ being a leaf in $H$ that,

[TABLE]

where $\alpha_{x,y}:=|S_{1}(x)\cap S_{1}(y)|$.

Remark 4.3**.**

Theorem 4.2 is an improvement of Theorem 4.1 for the case of $H$ being a star graph. Indeed, for any edge $e=x\sim y\in E_{0}=E(H)$ with $y$ being a leaf, we have

[TABLE]

In the above, we used $\alpha_{x,y}+2\leq|V(H)|$. We point out that, in case that $H$ is not a star graph, Theorem 4.1 can not be improved as $d_{x}d_{y}\kappa_{LLY}(e)\leq\max\{d_{x},d_{y}\}\left(\alpha_{x,y}+2+2|S_{1}(e)_{(H)}|\right)+2\min\{d_{x},d_{y}\}-2d_{x}d_{y}.$ A counterexample is given by an edge not included in a $3$-cycle of $P_{\infty}\times K_{3}$.

Proof.

Let $V(G)=X\sqcup Y$ be the partition of $V(G)$ such that $E(X,Y)=E_{0}=E(H)$ and $x\in X$. Set $Z:=X\setminus\{x\}$. For convenience, we denote $a:=\frac{1}{d_{x}(d_{y}+1)}$. By Kantorovich duality (2.1), we estimate

[TABLE]

Therefore, we deduce by (2.3) that

[TABLE]

This completes the proof. ∎

5. Proofs of the main results

We prove Theorem 1.3 and Theorem 1.4 in this section.

Proof of Theorem 1.3.

Let $E_{0}\subset E$ be a min-cut of $G$ and $H$ be the edge-induced subgraph on $E_{0}$ of $G$. Then $H$ is a bipartite graph without isolated vertices. We divide our arguments into two cases.

Case 1: $H$ is a star graph. Let $e=x\sim y\in E_{0}=E(H)$ be an edge of $H$. Observe that $|S_{1}(e)_{(H)}|=|E(H)|-1=k^{\prime}(G)-1$. We assume without loss of generality that $y$ is a leaf vertex in $H$. By Theorem 4.2, we derive that

[TABLE]

Since $E_{0}=E(H)$ is a min-cut, we have $d_{x}\geq 2k^{\prime}(G)$. Therefore, we deduce that

[TABLE]

Rearranging yields that

[TABLE]

Case 2: $H$ is not a star graph. Let $e=x\sim y\in E_{0}=E(H)$ be an edge of $H$ attaining the minimum of $|S_{1}(e)_{(H)}|$. We assume without loss of generality that $d_{x}\geq d_{y}$. Applying Theorem 4.1, we estimate

[TABLE]

In the second inequality above, we have applied Theorem 1.11 and the assumption $d_{y}\leq d_{x}$. Since $G$ has non-negative Lin–Lu–Yau curvature, we derive from (5.3) that

[TABLE]

This completes the proof of Theorem 1.3. ∎

Next, we prove Theorem 1.4. Let us first characterize possible local structures around the graph $H$ induced by a min-cut $E_{0}$.

Lemma 5.1**.**

Let $G=(V,E)$ be a connected graph with non-negative Lin–Lu–Yau curvature and $k^{\prime}(G)=\delta(G)-1$. Let $E_{0}\subset E$ be a min-cut of $G$ and $H$ be the edge-induced subgraph of $E_{0}$ in $G$. Denote by $V(G)=X\sqcup Y$ the partition of $V(G)$ such that $E(X,Y)=E_{0}=E(H)$. Set $A:=X\cap V(H)$ and $B:=Y\cap V(H)$.

• (1)

If $H$ is a star graph $K_{1,n}$ for $n\geq 1$, assuming without loss of generality that $A=\{x\}$ contains only one vertex $x$, then we have

• (a)

$d_{x}=2k^{\prime}(G)$;

• (b)

$d_{y}=k^{\prime}(G)+1$, for any $y\in B$;

• (c)

$G[B]=K_{|B|}$.

• (2)

If $H$ is not a star graph, then we have

• (a)

$H\in\{K_{2,2},\ nK_{1,1}\ \text{for }n\geq 2,\ 2K_{1,\frac{n}{2}}\ \text{for even }n>2,\ ST_{\frac{n-1}{2},\frac{n-1}{2}}\ \text{for odd }n\geq 3\}$;

• (b)

$d_{x}=d_{y}=k^{\prime}(G)+1$* for any $x\in A$ and $y\in B$;*

• (c)

$G[A]=K_{|A|}$* and $G[B]=K_{|B|}$.*

Proof.

We first assume that $H$ is a star graph. For any $y\in B$, we apply the argument of Case 1 in the proof of Theorem 1.3 to the edge $e=x\sim y\in E_{0}=E(H)$. Then the property $k^{\prime}(G)=\delta(G)-1$ forces the equalities in (5.2), that is, $d_{y}=k^{\prime}(G)+1=\alpha_{x,y}+2$. This proves $(1)(b)$ and $\alpha_{x,y}=k^{\prime}(G)-1=|E(H)|-1$. The latter shows that $y$ is adjacent to all other vertices in $B$. This proves (1)(c). Furthermore, the fact $k^{\prime}(G)=\delta(G)-1$ forces the equalities in (5.1), that is $\kappa(e)=0$ and $d_{x}=2k^{\prime}(G)$. The latter proves (1)(a).

Next, we assume that $H$ is not a star graph. Let $e=x\sim y\in E_{0}=E(H)$ be an edge assuming $\min_{e\in E(H)}|S_{1}(e)_{(H)}|$. Assume without loss of generality that $x\in X$ and $y\in Y$. We apply the argument of Case 2 in the proof of Theorem 1.3 to $e$. Then the property $k^{\prime}(G)=\delta(G)-1$ forces the equalities in (5.3) and (5.4). The first equality in (5.3) tells that $\kappa_{LLY}(e)=0$. Applying Theorem 4.1, we derive from the second equality that $x$ is adjacent to all other vertices in $A$ and $y$ is adjacent to all other vertices in $B$. The third equality in (5.3) forces $d_{x}=d_{y}$ and, by Theorem 1.11, (2)(a). The equalities in (5.4) imply that $d_{y}=k^{\prime}(G)+1$. Hence, we have $d_{x}=d_{y}=k^{\prime}(G)+1$.

If $H\in\{K_{2,2},\ nK_{1,1}\ \text{for }n\geq 2,\ 2K_{1,\frac{n}{2}}\ \text{for even }n>2\}$, we observe that every edge $e\in E(H)$ has the same value $|S_{1}(e)_{(H)}|$. Hence, we can apply the above argument for each $e\in E(H)$. This proves (2)(b) and (2)(c). If $H\not\in\{K_{2,2},\ nK_{1,1}\ \text{for }n\geq 2,\ 2K_{1,\frac{n}{2}}\ \text{for even }n>2\}$, we deduce by (2)(a) that $H=ST_{\frac{n-1}{2},\frac{n-1}{2}}$ for odd $n\geq 3$. Let $u\sim v,u\in A$ and $v\in B$ be the central edge of $ST_{\frac{n-1}{2},\frac{n-1}{2}}$. Then the edges assuming $\min_{e\in E(H)}|S_{1}(e)_{(H)}|$ are all edges in $E(H)$ except for $u\sim v$. Notice that those edges cover all vertices in $V(H)$. Applying the above argument for all those edges, we conclude (2)(b) and (2)(c). ∎

In the following lemma, we show that the local structure of the graph $H$ induced by a min-cut $E_{0}$ has further restrictions.

Lemma 5.2**.**

Let $G=(V,E)$ be a connected graph with non-negative Lin–Lu–Yau curvature and $k^{\prime}(G)=\delta(G)-1$. Let $E_{0}\subset E$ be a min-cut of $G$ and $H$ be the edge-induced subgraph of $E_{0}$ in $G$. Then we have

[TABLE]

Proof.

By Lemma 5.1, it is enough to show that $H$ cannot be $2K_{1,\frac{n}{2}}$ for even $n>4$ or $ST_{\frac{n-1}{2},\frac{n-1}{2}}$ for odd $n>3$. Denote by $V(G)=X\sqcup Y$ the partition of $V(G)$ such that $E(X,Y)=E_{0}=E(H)$. Set $A:=X\cap V(H)$ and $B:=Y\cap V(H)$. We argue by contradiction.

Case 1: Assume that $H=2K_{1,\frac{n}{2}}$ for even $n>4$. Then $n=k^{\prime}(G)$. By Lemma 5.1, we have either $G[A]=K_{2},G[B]=K_{n}$ or $G[A]=G[B]=K_{\frac{n}{2}+1}$, see Figure 8 and Figure 8.

We first consider the case that $G[A]=K_{2}$ and $G[B]=K_{n}$. Let us denote $A:=\{u_{1},u_{2}\}$ and $B:=\{v_{1},\ldots,v_{n}\}$ such that $u_{1}\sim v_{i}$ for any $i=1,\ldots\frac{n}{2}$ and $u_{2}\sim v_{j}$ for any $j=\frac{n}{2}+1,\ldots,n$. Applying Lemma 5.1, we have $d_{u_{i}}=d_{v_{j}}=k^{\prime}(G)+1=n+1$ for $i=1,2$ and $j=1,\dots,n$, $|S_{1}(u_{1})\setminus(A\cup B)|=\frac{n}{2}$ and $|S_{1}(v_{1})\setminus(A\cup B)|=1$. Denote $S_{1}(u_{1})\setminus(A\cup B)=\{w_{1},\dots,w_{\frac{n}{2}}\}$ and $S_{1}(v_{1})\setminus(A\cup B)=\{t\}$. Notice that

[TABLE]

Consider the Wasserstein distance $W(\mu_{u_{1}},\mu_{v_{1}})$, where $\mu_{u_{1}}:=\mu_{u_{1}}^{\frac{1}{n+2}},\mu_{v_{1}}:=\mu_{v_{1}}^{\frac{1}{n+2}}$ are the two uniform probability measures on $B_{1}(u_{1}),B_{1}(v_{1})$, respectively. By (2.4) and (2.5), an optimal transport plan can be induced by a bijective map $\phi:B_{1}(u_{1})\to B_{1}(v_{1})$ such that $\phi(u)=u$ for any $u\in B_{1}(u_{1})\cap B_{1}(v_{1})$. If at least two of the vertices $w_{1},\dots,w_{\frac{n}{2}}$, say $w_{1}$ and $w_{2}$, are not adjacent to $u_{2}$, then we have $d(w_{1},\phi(w_{1}))=d(w_{2},\phi(w_{2}))=3$, and hence

[TABLE]

By (2.3), this implies the Lin–Lu–Yau curvature $\kappa_{LLY}(u_{1},v_{1})<0$, a contradiction. Therefore, at most one of the vertices $w_{1},\dots,w_{\frac{n}{2}}$ is not adjacent to $u_{2}$. Without loss of generality, we assume $w_{i}\sim u_{2}$ for any $i=2,\ldots,\frac{n}{2}$. Since $d_{u_{2}}=k^{\prime}(G)+1=n+1$ by Lemma 5.1, the set $S_{1}(u_{2})\setminus(\{w_{2},\dots,w_{\frac{n}{2}}\}\cup A\cup B)$ contains only one vertex $w_{0}$. Notice that $w_{0}$ may coincide with $w_{1}$. Set $X^{\prime}:=X\setminus A$ and $Y^{\prime}:=Y\cup A$. Then $|E(X^{\prime},Y^{\prime})|=n$ and hence $E(X^{\prime},Y^{\prime})$ is a min-cut. Let $H^{\prime}$ be the edge-induced subgraph on $E(X^{\prime},Y^{\prime})$. Using Lemma 5.1 and recalling $n>4$ is even, we deduce that $H^{\prime}\in\{nK_{1,1},\ 2K_{1,\frac{n}{2}},K_{1,n}\}.$ Observe that $A^{\prime}:=X^{\prime}\cap V(H^{\prime})=\{w_{1},\dots,w_{\frac{n}{2}},w_{0}\}$ and $B^{\prime}:=Y^{\prime}\cap V(H^{\prime})=\{u_{1},u_{2}\}$. Then, $|A^{\prime}|$ equals either $n/2$ or $n/2+1$ and $|B^{\prime}|=2$. None of $nK_{1,1},\ 2K_{1,\frac{n}{2}},K_{1,n}$ satisfies this property. A contradiction. Therefore, the case $G[A]=K_{2}$ and $G[B]=K_{n}$ cannot happen.

Next, we consider the case that $G[A]=G[B]=K_{\frac{n}{2}+1}$. Denote $A=\{u_{1},\dots,u_{\frac{n}{2}+1}\}$ and $B=\{v_{1},\dots,v_{\frac{n}{2}+1}\}$, where $u_{1}\sim v_{i}$ for $i=1,\ldots,\frac{n}{2}$ and $u_{j}\sim v_{\frac{n}{2}+1}$ for $j=2,\dots,\frac{n}{2}+1$. By Lemma 5.1, we deduce that $d_{u_{i}}=d_{v_{i}}=k^{\prime}(G)+1=n+1$ for $i=1,\dots,\frac{n}{2}+1$, $|S_{1}(u_{1})\setminus(A\cup B)|=1$, and $|S_{1}(v_{1})\setminus(A\cup B)|=\frac{n}{2}$. Let $S_{1}(u_{1})\setminus(A\cup B)=\{t\}$ and $S_{1}(v_{1})\setminus(A\cup B)=\{w_{1},\dots,w_{\frac{n}{2}}\}$. Then

[TABLE]

Let $\phi:B_{1}(u_{1})\to B_{1}(v_{1})$ be a bijective map satisfying $\phi(u)=u$ for any $u\in B_{1}(u_{1})\cap B_{1}(v_{1})$, which induces an optimal transport plan between $\mu_{u_{1}}$ and $\mu_{v_{1}}$. If at least two of the vertices $w_{1},\dots,w_{\frac{n}{2}}$, say $w_{1}$ and $w_{2}$, are not adjacent to $v_{\frac{n}{2}+1}$, then $d(\phi^{-1}(w_{1}),w_{1})=d(\phi^{-1}(w_{2}),w_{2})=3$. Thus,

[TABLE]

By (2.3), we have $\kappa_{LLY}(u_{1},v_{1})<0$, a contradiction. Therefore, at most one of the vertices $w_{1},\dots,w_{\frac{r}{2}}$ is not adjacent to $v_{\frac{r}{2}+1}$. Without loss of generality, we assume let $w_{i}\sim v_{\frac{n}{2}+1}$ for $i=2,\dots,\frac{n}{2}$. Then

[TABLE]

However, we have $d_{v_{\frac{n}{2}+1}}=k^{\prime}(G)+1=n+1$ by Lemma 5.1. Since $n>4$ is even, we have $\frac{3n}{2}-1>n+1$, a contradiction. Therefore, the case that $G[A]=G[B]=K_{\frac{n}{2}+1}$ cannot happen.

In conclusion, we have $H$ cannot be $2K_{1,\frac{n}{2}}$ for even $n>4$.

Case 2: Assume that $H=ST_{\frac{n-1}{2},\frac{n-1}{2}}$ for odd $n>3$. Then $n=k^{\prime}(G)$. Denote $A=\{u_{1},\dots,u_{\frac{n+1}{2}}\}$ and $B=\{v_{1},\dots,v_{\frac{n+1}{2}}\}$, where $u_{1}\sim v_{i}$ and $u_{i}\sim v_{\frac{n+1}{2}}$ for $i=1,\dots,\frac{n+1}{2}$. By Lemma 5.1, we deduce that $d_{u_{i}}=d_{v_{i}}=k^{\prime}(G)+1=n+1$ for $i=1,\dots,\frac{n+1}{2}$, $|S_{1}(u_{1})\setminus(A\cup B)|=1$ and $|S_{1}(v_{1})\setminus(A\cup B)|=\frac{n+1}{2}$. Let $S_{1}(u_{1})\setminus(A\cup B)=\{t\}$ and $S_{1}(v_{1})\setminus(A\cup B)=\{w_{1},\dots,w_{\frac{n+1}{2}}\}$. Then

[TABLE]

Let $\phi:B_{1}(u_{1})\to B_{1}(v_{1})$ be a bijective map satisfying $\phi(u)=u$ for any $u\in B_{1}(u_{1})\cap B_{1}(v_{1})$, which induces an optimal transport plan between $\mu_{u_{1}}$ and $\mu_{v_{1}}$. If at least two of the vertices $w_{1},\dots,w_{\frac{n+1}{2}}$, say $w_{1}$ and $w_{2}$, are not adjacent to $v_{\frac{n+1}{2}}$, then $d(\phi^{-1}(w_{1}),w_{1})=d(\phi^{-1}(w_{2}),w_{2})=3$. Thus,

[TABLE]

By (2.3), we have $\kappa_{LLY}(u_{1},v_{1})<0$, a contradiction. Therefore, at most one of the vertices $w_{1},\dots,w_{\frac{n+1}{2}}$ is not adjacent to $v_{\frac{n+1}{2}}$. Without loss of generality, we assume that $w_{i}\sim v_{\frac{r+1}{2}}$ for any $i=2,\dots,\frac{n+1}{2}$. Then

[TABLE]

Lemma 5.1 tells that $d_{v_{\frac{n+1}{2}}}=n+1$. However, $\frac{3n-1}{2}>n+1$ for odd $n>3$. This is a contradiction. Therefore, $H$ cannot be $ST_{\frac{n-1}{2},\frac{n-1}{2}}$ for any odd $n>3$. This finishes the proof. ∎

Now we are prepared for the proof of Theorem 1.4.

Proof of Theorem 1.4.

Let $G$ be a connected graph with non-negative Lin–Lu–Yau curvature and $k^{\prime}(G)=\delta(G)-1$. Let $E_{0}\subset E$ be a min-cut of $G$. Denote by $H_{0}$ the edge-induced subgraph of $E_{0}$ in $G$, and $V(G)=X_{0}\sqcup Y_{0}$ be the partition of $V(G)$ such that $E(X_{0},Y_{0})=E_{0}=E(H_{0})$. Set $A_{0}=X_{0}\cap V(H_{0})$, $B_{0}=Y_{0}\cap V(H_{0})$. Denote by $n:=k^{\prime}(G)$ for convenience.

By Lemma 5.2, we have

[TABLE]

Note that, if $H_{0}=2K_{1,2}$, we have either $|A_{0}|=|B_{0}|=3$ or $|A_{0}|,|B_{0}|\in\{2,4\}$. We will discuss the case that $H_{0}=2K_{1,2}$ with $|A_{0}|=|B_{0}|=3$ separately, and deal with the remaining cases first.

Case 1: Either $H_{0}\neq 2K_{1,2}$, or $H_{0}=2K_{1,2}$ with $|A_{0}|,|B_{0}|\in\{2,4\}$.

Set $V_{0}=A_{0}$ and $V_{1}=B_{0}$. Then we have $V_{0}\cap V_{1}=\emptyset$. By Lemma 5.1, it holds that $G[V_{0}]=K_{|V_{0}|}$ and $G[V_{1}]=K_{|V_{1}|}$. Let us define for $i\geq 2,i\in\mathbb{Z}$ that

[TABLE]

and for $i\leq-1,i\in\mathbb{Z}$ that

[TABLE]

Claim 5.3**.**

For any $i\in\mathbb{Z}$, the edge set $E(V_{i-1},V_{i})$ is a min-cut of $G$ and the subgraph $G[V_{i-1},V_{i}]$ coincides with the edge-induced graph of $E(V_{i-1},V_{i})$.

Proof.

We prove by induction. The claim holds true for $i=1$. For any $k\geq 1$, suppose it is true for $i\leq k$. Let $V(G)=X\sqcup Y$ be the partition of $V(G)$ such that $V_{k-1}\subset X$, $V_{k}\subset Y$, and $E(X,Y)=E(V_{k-1},V_{k})$. Then the sets $X^{\prime}:=X\cup V_{k}$ and $Y^{\prime}:=Y\setminus V_{k}$ provide a new partition of $V(G)$ such that $E(X^{\prime},Y^{\prime})=E(V_{k},V_{k+1})$. That is, $E(V_{k},V_{k+1})$ is a cut. Next, we show that $|E(V_{k},V_{k+1})|=n:=k^{\prime}(G)$ and hence $E(V_{k},V_{k+1})$ is a min-cut. Observe that

[TABLE]

By the induction assumption, $E(V_{k-1},V_{k})$ is a min-cut of $G$ and the subgraph $G[V_{k-1},V_{k}]$ coincides with the edge-induced graph of $E(V_{k-1},V_{k})$. By Lemma 5.1, we have $G[V_{k}]=K_{|V_{k}|}$, and for any $x\in V_{k}$ that

[TABLE]

If $|V_{k}|=1$, then we deduce from (5.5) that $E(V_{k-1},V_{k})=K_{1,n}$. Denote $V_{k}=\{u\}$. Then $|S_{1}(u)\cap(V_{k-1}\cup V_{k})|=|S_{1}(u)\cap V_{k-1}|=n$. By (5.6), this yields that $|E(V_{k},V_{k+1})|=2n-n=n$. By the definition of $V_{k+1}$, the subgraph $G[V_{k},V_{k+1}]$ coincides with the edge-induced graph of $E(V_{k},V_{k+1})$.

If, otherwise, $|V_{k}|>1$, then we have by (5.5) that $E(V_{k-1},V_{k})\in\{nK_{1,1},ST_{1,1},K_{2,2},2K_{1,2}\}$. If $E(V_{i-1},V_{i})=nK_{1,1}$, then $G[V_{k}]=K_{|V_{k}|}=K_{n}$. Moreover, for each $x\in V_{k}$,

[TABLE]

Applying (5.6), we obtain $|E(V_{k},V_{k+1})|=n\times[(n+1)-n]=n$. If $E(V_{i-1},V_{i})=ST_{1,1}$, then $n=3$ and $G[V_{k}]=K_{2}$. Denoting $V_{k}=\{u_{1},u_{2}\}$, we have $d_{u_{1}}=d_{u_{2}}=4$, and

[TABLE]

Hence, by (5.6), $|E(V_{k},V_{k+1})|=d_{u_{1}}+d_{u_{2}}-5=3=n$. If $E(V_{k-1},V_{k})=K_{2,2}$, then $n=4$ and $G[V_{k}]=K_{2}$. For every $x\in V_{k}$,

[TABLE]

Therefore, $|E(V_{k},V_{k+1})|=2\times(5-3)=4=n$. Finally, suppose that $E(V_{k-1},V_{k})=2K_{1,2}$, then $n=4$. We claim that that $|V_{k}|=2$ or $4$. When $k=1$, this follows directly from the assumption in Case 1. For $k>1$, if this were not true, then $|V_{k-1}|=|V_{k}|=3$. By Lemma 5.1, it follows that $G[V_{k-1}]=K_{3}$ and each vertex $u\in V_{k-1}$ has degree $5$. Hence, $|E(V_{k-2},V_{k-1]})|=15-4-6=5>4$, contradicting the assumption that $E(V_{k-2},V_{k-1})$ is a min-cut.

If $|V_{k}|=2$, then $G[V_{k}]=K_{2}$ and for any $x\in V_{k}$, we have $|S_{1}(x)\cap(V_{k-1}\cup V_{k})|=2+1=3$. If $|V_{k}|=4$, then $G[V_{k}]=K_{4}$ and for each $x\in V_{k}$, $|S_{1}(x)\cap(V_{k-1}\cup V_{k})|=1+3=4$. Consequently, in either situation, $|E(V_{k},V_{k+1})|=4=n$.

The above analysis shows that $S_{1}(x)\cap V_{k+1}\neq\emptyset$ for any $x\in V_{k}$. Combined with the definition of $V_{k+1}$, this implies that the subgraph $G[V_{k},V_{k+1}]$ coincides with the edge-induced graph of $E(V_{k-1},V_{k})$.

By the above induction argument, we show Claim 5.3 for any $i\geq 1$. The case $i\leq 1$ follows from a similar induction argument. This completes the proof. ∎

By definition, for every $i\in\mathbb{Z}$, we have $V_{i}\cap V_{i+1}=\emptyset$. Moreover, since $E(V_{i},V_{i+1})$ is a min-cut for each $i\in\mathbb{Z}$, it follows that $V_{i}\cap V_{j}=\emptyset$ and $E(V_{i},V_{j})=\emptyset$ whenever $|i-j|>1$. Because $G$ is connected, it follows that $V(G)=\sqcup_{i\in\mathbb{Z}}V_{i}$ with $E(V_{i},V_{j})=\emptyset$ whenever $|i-j|>1$.

We examine the following three subcases according to the value of $n=k^{\prime}(G)$.

Subcaes 1.1: $n=1,2$ or $n\geq 5$.

By Lemma 5.1, Lemma 5.2, and Claim 5.3, for any $i\in\mathbb{Z}$, we have $G[V_{i}]\in\{K_{1},K_{n}\}$ and $G[V_{i},V_{i+1}]\in\{K_{1,n},nK_{1,1}\}$. Therefore, $G\in\mathcal{G}_{n}$.

Subcaes 1.2: $n=3$.

If $E(V_{i},V_{i+1})\neq ST_{1,1}$ for any $i\in\mathbb{Z}$, then, by Lemma 5.1, Lemma 5.2, and Claim 5.3, we have $G[V_{i}]\in\{K_{1},K_{3}\}$ and $G[V_{i},V_{i+1}]\in\{K_{1,3},3K_{1,1}\}$ for any $i\in\mathbb{Z}$. Therefore, $G\in\mathcal{G}_{3}$.

If, otherwise, there exists an $i_{0}\in\mathbb{Z}$ such that $E(V_{i_{0}},V_{i_{0}+1})=ST_{1,1}$, then we claim that $E(V_{i-1},V_{i})=ST_{1,1}$ for any $i\in\mathbb{Z}$. Without loss of generality, suppose $E(V_{0},V_{1})=ST_{1,1}$. We argue by induction. The claim holds true for $i=1$. Suppose it is true for $i=k\geq 1$. By Lemma 5.1 and Lemma 5.2, we have $E(V_{k},V_{k+1})\in\{K_{1,3},3K_{1,1},ST_{1,1}\}$. Since $|V_{k}|=2$, $E(V_{k},V_{k+1})=ST_{1,1}$. This shows the claim for any $i\geq 1$. The case $i\leq 1$ follows by a similar induction argument.

Furthermore, applying Lemma 5.1 tells that $G$ is $4-$regular and $G[V_{i}]=K_{2}$ for any $i\in\mathbb{Z}$. Therefore, $G=G_{3}^{*}$.

Subcaes 1.3: $n=4$.

By Lemma 5.1, Lemma 5.2, and Claim 5.3, we have $G[V_{i}]\in\{K_{1},K_{2},K_{4}\}$ for any $i\in\mathbb{Z}$, and $G[V_{i},V_{i+1}]\in\{K_{1,4},4K_{1,1},K_{2,2},2K_{1,2}\}$ for any $i\in\mathbb{Z}$. Suppose that there exists $i\in\mathbb{Z}$ such that $(|V_{i-1}|,|V_{i}|,|V_{i+1}|)=(4,2,4)$, then the local structure of the graph is as depicted in Figure 4. It is direct to check by definition that the Lin–Lu–Yau curvature of any edge in $E(V_{i-1},V_{i})\sqcup E(V_{i},V_{i+1})$ is equal to $-0.2$. This contradicts to the assumption that $G$ has non-negative Lin–Lu–Yau curvature. Therefore, $G\in\tilde{\mathcal{G}}_{4}$.

To conclude the proof for Case 1, we mention that all graphs in $\mathcal{G}_{n},n\in\mathbb{Z}$ and $\tilde{\mathcal{G}}_{4}$ and the graph $G_{3}^{*}$ have non-negative Lin–Lu–Yau curvature. See Appendix A.

Case 2: $H_{0}=2K_{1,2}$ with $|A_{0}|=|B_{0}|=3$.

In this case, we have $n=|H_{0}|=4$, $\delta(G)=n+1=5$. Let $A_{0}=\{u_{1},u_{2},u_{3}\}$ and $B_{0}=\{v_{1},v_{2},v_{3}\}$, where $u_{1}\sim v_{i}$ and $u_{1+i}\sim v_{3}$ for $i=1,2$. By Lemma 5.1, we have $G[A_{0}]=G[B_{0}]=K_{3}$, and $d_{u_{i}}=d_{v_{i}}=n+1=5$ for $i=1,2,3$. Therefore, $|S_{1}(u_{1})\setminus(A_{0}\cup B_{0})|=|S_{1}(v_{3})\setminus(A_{0}\cup B_{0})|=1$. Let $S_{1}(u_{1})\setminus(A_{0}\cup B_{0})=\{u_{4}\}$ and $S_{1}(v_{3})\setminus(A_{0}\cup B_{0})=\{v_{4}\}$. Set $V_{0}=A_{0}\cup\{u_{4}\}$ and $V_{1}=B_{0}\cup\{v_{4}\}$.

Claim 5.4**.**

$G[V_{0}]=G[V_{1}]=K_{4}$ and $d_{u_{4}}=d_{v_{4}}=5$.

Proof.

Let us consider the Lin–Lu–Yau curvature of the edge $e_{1}:=u_{1}\sim v_{1}$. Notice that

[TABLE]

Observe that $|S_{1}(v_{1})\setminus(A_{0}\cup B_{0})|=5-3=2$. Denote $S_{1}(v_{1})\setminus(A_{0}\cup B_{0})=\{w_{1},w_{2}\}$, see Figure 9. Then

[TABLE]

Recall from (2.4) that there exists a bijective map $\phi:B_{1}(u_{1})\longrightarrow B_{1}(v_{1})$ such that $\phi(u)=u$ for any $u\in B_{1}(u_{1})\cap B_{1}(v_{1})$ and

[TABLE]

If $v_{3}\not\sim w_{1}$ and $v_{3}\not\sim w_{2}$, then $d(\phi^{-1}(w_{1}),w_{1})=d(\phi^{-1}(w_{2}),w_{2})=3$. Thus,

[TABLE]

By (2.3), the above inequality implies that $\kappa_{LLY}(e_{1})<0$, a contradiction. Therefore, we have $v_{3}\sim w_{1}$ or $v_{3}\sim w_{2}$. Without loss of generality, let $v_{3}\sim w_{2}$. Since $d_{v_{3}}=5$, we have $v_{4}=w_{2}$ and hence $v_{4}\sim v_{1}$. By a similar argument for the edge $u_{1}\sim v_{2}$, we obtain $v_{4}\sim v_{2}$. Hence, $G[V_{1}]=K_{4}$.

Similarly, we derive $u_{4}\sim u_{2}$, $u_{4}\sim u_{3}$, and hence $G[V_{0}]=K_{4}$. Next, we show $d_{u_{4}}=d_{v_{4}}=5$. Since $\delta(G)=5$, $d_{u_{4}},d_{v_{4}}\geq 5$.

Suppose $d_{u_{4}}\geq 6$. We consider the Lin–Lu–Yau curvature of the edge $e_{*}=u_{1}\sim u_{4}$. Set $T:=S_{1}(u_{4})\setminus\{u_{1},u_{2},u_{3}\}$, see Figure 10. By the Kantorovich duality (2.1), we have

[TABLE]

Since $d(v,T)=0$ for any $v\in T$, we have

[TABLE]

By (2.3), the above inequality implies that $\kappa_{LLY}(e_{*})<0$, a contradiction. Thus, we derive $d_{u_{4}}=5$. Similarly, we have $d_{v_{4}}=5$.

∎

Let us define, for $i\geq 2,i\in\mathbb{Z}$, that $V_{i}:=R_{i}^{1}\sqcup R_{i}^{2}$, where

[TABLE]

and for $i\leq-1,i\in\mathbb{Z}$, that $V_{i}=L_{i}^{1}\sqcup L_{i}^{2}$, where

[TABLE]

Observe from the above definition that $V_{i}\cap V_{i+1}=\emptyset$ for any $i\in\mathbb{Z}$. For any $i\in\mathbb{Z}$, we denote by $H_{i}$ the edge-induced subgraph of $E(V_{i},V_{i+1})$ in $G$. Set $A_{i}:=V_{i}\cap V(H_{i})$ and $B_{i}:=V_{i+1}\cap V(H_{i})$.

Next, we show for each $i\in\mathbb{Z}$, the following properties hold true:

1. (1)

$E(V_{i-1},V_{i})$ is a min-cut of $G$, with $H_{i-1}=2K_{1,2}$ and $|A_{i-1}|=|B_{i-1}|=3$; 2. (2)

$d_{x}=5$ for any $x\in V_{i-1}\cup V_{i}$; 3. (3)

$G[V_{i}]=K_{4}$.

We again argue by induction. By Claim 5.4, The above properties hold true for $i=1$. Suppose they are true for $i=k\geq 1$. Let $V(G)=X\sqcup Y$ be the partition of $V(G)$ such that $V_{k-1}\subset X$, $V_{k}\subset Y$, and $E(X,Y)=E(V_{k-1},V_{k})$. Since $E(V_{k-1},V_{k})$ is a cut of $G$, we have $V_{k+1}\cap X=\emptyset$. Then the sets $X^{\prime}:=X\cup V_{k}$ and $Y^{\prime}:=Y\setminus V_{k}$ provide a new partition of $V(G)$ such that $E(X^{\prime},Y^{\prime})=E(V_{k},V_{k+1})$. Thus, $E(V_{k},V_{k+1})$ is a cut. Next, we show it is a min-cut.

Let $V_{j}:=\{u_{j}^{1},u_{j}^{2},u_{j}^{3},u_{j}^{4}\}$ for $j=k-1,k$. Without loss of generality, we assume that

[TABLE]

By the induction assumption, $d_{x}=5$ for any $x\in V_{k}$. Set $S_{1}(u_{k}^{4})\setminus\{u_{k}^{1},u_{k}^{2},u_{k}^{3}\}:=\{u_{k+1}^{1},u_{k+1}^{2}\}$, $S_{1}(u_{k}^{2})\setminus\{u_{k-1}^{1},u_{k}^{1},u_{k}^{3},u_{k}^{4}\}:=\{u_{k+1}^{3}\}$, and $S_{1}(u_{k}^{1})\setminus\{u_{k-1}^{1},u_{k}^{2},u_{k}^{3},u_{k}^{4}\}:=\{\tilde{u}_{k+1}^{3}\}$. Notice further that all neighbors of $u_{k}^{3}$ lie in $V_{k-1}\sqcup V_{k}$. Thus, $|E(V_{k},V_{k+1})|=4$ and $E(V_{k},V_{k+1})$ is a min-cut. Recall that $A_{k}=V(H_{k})\cap V_{k}=\{u_{k}^{1},u_{k}^{2},u_{k}^{4}\}$. Then, by Lemma 5.2, we have $H_{k}=2K_{1,2}$ with $|A_{k}|=|B_{k}|=3$. This proves the property $(1)$ holds for $i=k+1$.

Moreover, we conclude that $u_{k+1}^{3}=\tilde{u}_{k+1}^{3}$. Thus, $B_{k}=V(H_{k})\cap V_{k+1}=\{u_{k+1}^{1},u_{k+1}^{2},u_{k+1}^{3}\}$. By Lemma 5.1, we have $G[B_{k}]=K_{3}$, and $d_{u_{k+1}^{j}}=5$ for $j=1,2,3$. Let $S_{1}(u_{k+1}^{3})\setminus\{u_{k}^{1},u_{k}^{2},u_{k+1}^{1},u_{k+1}^{2}\}=\{u_{k+1}^{4}\}$. By Claim 5.4, we have $G[\{u_{k+1}^{1},u_{k+1}^{2},u_{k+1}^{3},u_{k+1}^{4}\}]=K_{4}$ and $d_{u_{k+1}^{4}}=5$. According to the definition of $V_{k+1}$, we have $V_{k+1}=\{u_{k+1}^{1},u_{k+1}^{2},u_{k+1}^{3},u_{k+1}^{4}\}$. Thus, $G[V_{k+1}]=K_{4}$, and $d_{x}=5$ for any $x\in V_{k}\cup V_{k+1}$. That is, the properties $(2)$ and $(3)$ hold for $i=k+1$. By our induction argument, we show the properties $(1)-(3)$ hold for $i\geq 1$. The case $i\leq 1$ follows by a similar induction argument.

Recall that $V_{i}\cap V_{i+1}=\emptyset$ for any $i\in\mathbb{Z}$. Since $E(V_{i},V_{i+1})$ is a min-cut for each $i\in\mathbb{Z}$, we observe that $V_{i}\cap V_{j}=\emptyset$ and $E(V_{i},V_{j})=\emptyset$ whenever $|i-j|>1$. Since $G$ is connected, we have $V(G)=\sqcup_{i\in\mathbb{Z}}V_{i}$. Therefore, $G=G_{4}^{*}$. Moreover, notice that the graph $G_{4}^{*}$ has non-negative Lin–Lu–Yau curvature, see Appendix A.

Combining the above discussions in Case 1 and Case 2, we finish the proof of Theorem 1.4. ∎

Appendix A The Lin–Lu–Yau curvature of graphs listed in Theorem 1.4

In this Appendix, we present the Lin–Lu–Yau curvature values for edges of graphs $G$ in

[TABLE]

The Lin–Lu–Yau curvature of an edge is determined using (2.3). Notice that the Wasserstein distance $W$ can be obtained by using Definition 2.1 together with the Kantorovich duality formula (2.1). More precisely, Definition 2.1 provides an upper bound for $W$, while the Kantorovich duality (2.1) gives a lower bound. When these two bounds agree, the exact value of $W$ follows. One may also use the graph curvature calculator (https://www.mas.ncl.ac.uk/graph-curvature/) to compute Lin–Lu–Yau curvature values, see [9].

Case 1: $G=G_{3}^{*}$. Let $V(G)=\sqcup_{i\in\mathbb{Z}}V_{i}$ be the partition in Definition 1.7. Then an edge $e$ of $G$ lies either in $G[V_{i}]$ or in $E(V_{i},V_{i+1})$ for some integer $i$. If $e\in E(G[V_{i}])$, then $\kappa_{LLY}(e)=0.5$, see Figure 12. If, otherwise, $e\in E(V_{i},V_{i+1})$, then we have $\kappa_{LLY}(e)\in\{0,0.5\}$, see Figure 12.

Case 2: $G=G_{4}^{*}$. Let $V(G)=\sqcup_{i\in\mathbb{Z}}V_{i}$ be the partition in Definition 1.8. Then an edge $e$ of $G$ lies either in $G[V_{i}]$ or in $E(V_{i},V_{i+1})$ for some integer $i$. If $e\in E(G[V_{i}])$, then $\kappa_{LLY}(e)\in\{0,0.4,1.2\}$, see Figure 14. If, otherwise, $e\in E(V_{i},V_{i+1})$, then we have $\kappa_{LLY}(e)=0$, see Figure 14.

Case 3: $G\in\mathcal{G}_{n}$ for some positive integer $n$. Let $V(G)=\sqcup_{i\in\mathbb{Z}}V_{i}$ be the partition in Definition 1.6. Then an edge $e$ of $G$ lies either in $G[V_{i}]$ or in $E(V_{i},V_{i+1})$ for some integer $i$. If $e\in E(G[V_{i}])$, then $G[V_{i}]$ is not $K_{1}$ but $K_{n}$, and the local structure around $e$ can only be one of the following three subcases: (i) $G[V_{i-1}]=G[V_{i+1}]=K_{1}$; (ii) $\{G[V_{i-1}],G[V_{i+1}]\}=\{K_{1},K_{n}\}$; (iii) $G[V_{i-1}]=G[V_{i+1}]=K_{n}$; see Figure 16, 16, and 17. The curvature $\kappa_{LLY}(e)$ equals $\frac{n+2}{n+1}$, $1$, $\frac{n}{n+1}$, respectively.

If, otherwise, $e\in E(V_{i},V_{i+1})$, then we have $\kappa_{LLY}(e)=0$. In fact, the edge-induced subgraph of $E(V_{i},V_{i+1})$ is isomorphic to either $K_{1,n}$ or $nK_{1,1}$, see Figure 19 and 19.

Case 4: $G\in\tilde{\mathcal{G}}_{4}$. Let $V(G)=\sqcup_{i\in\mathbb{Z}}V_{i}$ be the partition in Definition 1.9. Then an edge $e$ of $G$ lies either in $G[V_{i}]$ or in $E(V_{i},V_{i+1})$ for some integer $i$. If $e\in E(G[V_{i}])$, then $G[V_{i}]\in\{K_{2},K_{4}\}$. If $G[V_{i}]=K_{4}$, then the local structure near $e$ can only be one of the following six subcases:

1. (1)

$G[V_{i-1}]=G[V_{i+1}]=K_{1}$; 2. (2)

$G[V_{i-1}]=G[V_{i+1}]=K_{2}$; 3. (3)

$G[V_{i-1}]=G[V_{i+1}]=K_{4}$; 4. (4)

$\{G[V_{i-1}],G[V_{i+1}]\}=\{K_{1},K_{2}\}$; 5. (5)

$\{G[V_{i-1}],G[V_{i+1}]\}=\{K_{1},K_{4}\}$; 6. (6)

$\{G[V_{i-1}],G[V_{i+1}]\}=\{K_{2},K_{4}\}$.

The curvature value of $e$ is depicted in Figure 21–25.

If $G[V_{i}]=K_{2}$, then the local structure around $e$ can only be one of the following two subcases: (a) $G[V_{i-1}]=G[V_{i+1}]=K_{2}$; (b) $\{G[V_{i-1}],G[V_{i+1}]\}=\{K_{2},K_{4}\}$. See Figures 27 and Figure 27 for the curvature values.

If $e\in E(V_{i},V_{i+1})$, then we have $\kappa_{LLY}(e)=0$. In fact, the edge-induced subgraph of $E(V_{i},V_{i+1})$ is isomorphic to one of $K_{1,4}$, $4K_{1,1}$, and $2K_{1,2}$, see Figure 30, 30, and 30. Notice that in Figure 30, the induced subgraph $G[V_{i-1}]$ can only be $K_{2}$ due to Condition (iv) in Definition 1.9.

Acknowledgements

This work is supported by the National Key R & D Program of China 2023YFA1010200 and the National Natural Science Foundation of China No. 12431004. We are very grateful to Kaizhe Chen for many useful discussions, especially for providing one of the sharp graphs, $G_{3}^{*}$.