arXiv subjects
Publications and source records attributed to Shuliang Bai.
We study the continuous Lin--Lu--Yau Ricci flow on a finite tree in the inverse-weight case. We investigate the diffusive structure of the curvature evolution equation and prove the convergence of the curvature along the Ricci flow. Moreover, we show that, in logarithmic coordinates, the Ricci flow can be characterized as the gradient flow of a convex potential.
Let $G$ be a finite simple graph with at least one edge. We prove the sharp vanishing theorem \[ \kappa_{\min}^{\mathrm{LLY}}(G)>\frac12 \quad\Longrightarrow\quad \PathH_1(G;\R)=0. \] Equivalently, nonzero first GLMY path homology forces an edge of Lin--Lu--Yau curvature at most $1/2$. The threshold $1/2$ is sharp and is attained by $C_5$. The proof combines the cycle-space description of first GLMY path homology with the limit-free Laplacian characterization of Lin--Lu--Yau curvature. As a secondary consequence of the curvature-preserving universal-cover method, we prove that if $G$ is connected and $\kappa_{\min}^{\mathrm{LLY}}(G)>0$, then $\pi_1(\Xshort{5}(G),o)$ is finite, where $\Xshort{5}(G)$ is obtained by filling every simple cycle of length at most five. Equivalently, the normal subgroup generated by based simple $5$-cycle loops has finite index in $\pi_1^{\mathrm{GLMY}}(G,o)$. In higher degrees the situation is different: for each integer $r\geq1$, the Cartesian product $T_r=C_5^{\square r}$ has curvature $1/(2r)$ on every edge and, for every field $\F$, \[ \PathH_p(T_r;\F)\cong\F^{\binom rp}\qquad(0\leq p\leq r), \] so strict positivity of Lin--Lu--Yau curvature does not force higher-dimensional GLMY path homology to vanish.
In earlier work with Cheng and Hua we showed that on a finite tree the discrete Einstein metrics of the Lin--Lu--Yau curvature are the Perron eigenvector of an edge-indexed Ricci matrix. We extend this theory to unicyclic graphs. We determine exactly when the tree picture persists -- the balanced regime, where the spectrum becomes periodic rather than Dirichlet-type -- and compute it in closed form for bare cycles and for regular suns (cycles with pendant leaves); for a single decorated vertex on a long cycle it persists up to an explicit golden-ratio threshold. Beyond this regime the problem is piecewise-linear, and phenomena impossible on a tree appear: the Einstein metric can be non-unique, or absent -- a triangle with a pendant leaf carries none. For the regular suns we prove that it exists and is unique.
The Ricci matrix $R_T$ of a finite tree encodes its discrete Einstein metrics via the Perron eigenvector, with Lin-Lu-Yau's Ollivier Ricci curvature: $\kappa = -\lambda_{\max}(R_T)$. We show that edge subdivision, the natural operation of lengthening a tree, can decrease, preserve, or increase $\lambda_{\max}$. Compressing each branch into a scalar feedback function via the Schur complement reduces the spectral problem to a one-dimensional Chebyshev equation. We obtain an exact one-step trichotomy, a scalar transmission equation for arbitrary length, and the long-chain limit. Examples on double stars, including an asymmetric case where subdivision strictly increases $\lambda_{\max}$, illustrate the theory.
Let $R_T$ be the Ricci matrix of a finite tree $T$ introduced in \cite{BaiChengHua2026}, the largest eigenvalue $\lambda_{\max}(R_T)$ determines the sign of a discrete Einstein metric curvature on the tree. This paper investigates the asymptotic behavior of the sequence $\lambda_k = \lambda_{\max}(R_{T_k})$ obtained by repeatedly adding pendant edges at a fixed vertex. We prove that $\lambda_k$ converges to a limit $\lambda_\infty$ that depends only on the local branch data of $T$, and establish a first-order asymptotic expansion: \[ \lambda_k = \lambda_\infty + \frac{\alpha}{d+k} + O\!\left(\frac{1}{(d+k)^2}\right), \] where $d$ is the degree of the original vertex, and the coefficient $\alpha$ is given by a spectral projection. As a corollary, when $\alpha \neq 0$, $\lambda_k$ is eventually strictly monotonic (increasing or decreasing). This theory reveals the fine influence of local leaf addition on the global spectrum.
We establish the existence and uniqueness of discrete Einstein metrics on trees under Lin-Lu-Yau Ricci curvature using Perron-Frobenius theory. We establish a sharp upper bound for the largest eigenvalue of the associated Ricci matrix in terms of the maximum degree. Turning to structural properties, notably, the existence of a positive-curvature Einstein metric implies the tree must be a caterpillar. Furthermore, these metrics exhibit radial monotonicity, with edge weights decreasing strictly away from the maximal edge.
In this paper, we propose a type of Ricci flow on graphs where the probability distribution for the Lin-Lu-Yau curvature remains constant over time, and also study the related Forman curvature flow. These two curvature flows coincide on trees. We first prove the existence and uniqueness of solutions for both curvature flows in general graphs. Then, we obtain that the normalized curvature flow on trees converges to a constant curvature metric, and under the uniform measure, a complete classification of trees can be obtained based on the convergence results.
In this paper, we study the evolution of metrics on finite trees under continuous-time Ricci flows based on the Lin-Lu-Yau version of Ollivier Ricci curvature. We analyze long-time dynamics of edge weights and curvatures, providing precise characterizations of their limiting behaviors. We prove that the Ricci flow converges to metric with zero curvature on edges whose normalized weights converge to positive values only if the tree is a caterpillar tree.
Ricci curvature and Ricci flow have proven to be powerful tools for analyzing the geometry of discrete structures, particularly on undirected graphs, where they have been applied to tasks ranging from community detection to graph representation learning. However, their development on directed graphs remains limited, with Ricci flow being especially underexplored. In this work, we introduce a rigorous formulation of Ricci flow on directed weighted graphs, which evolves edge weights while preserving distances, and establish both the existence and uniqueness of its solutions. To capture the essence of asymmetry in directed networks and to enhance the capability of modeling more flexible structures, we incorporate a node-wise balancing factor that regulates between outflow and inflow. Building on the continuous Ricci flow evolution framework, we propose a discrete Ricci flow algorithm that is applicable to numerical computing. Numerical studies on various directed graph examples demonstrate the capacity of the proposed flow to reveal structural asymmetry and dynamic evolutions.
A graph is called Ricci-flat if its Ricci curvatures vanish on all edges, here the definition of Ricci curvature on graphs was given by Lin-Lu-Yau. The authors in arXiv:1301.0102 and arXiv:1802.02982 obtained a complete characterization for all Ricci-flat graphs with girth at least five. In this paper, we completely determined all Ricci-flat graphs with maximum degree at most 4.
We consider the Turán problems of $2$-edge-colored graphs. A $2$-edge-colored graph $H=(V, E_r, E_b)$ is a triple consisting of the vertex set $V$, the set of red edges $E_r$ and the set of blue edges $E_b$ with $E_r$ and $E_b$ do not have to be disjoint. The Turán density $π(H)$ of $H$ is defined to be $\lim_{n\to\infty} \max_{G_n}h_n(G_n)$, where $G_n$ is chosen among all possible $2$-edge-colored graphs on $n$ vertices containing no $H$ as a sub-graph and $h_n(G_n)=(|E_r(G)|+|E_b(G)|)/{n\choose 2}$ is the formula to measure the edge density of $G_n$. We will determine the Turán densities of all $2$-edge-colored bipartite graphs. We also give an important application of our study on the Turán problems of $\{2, 3\}$-hypergraphs.
In this paper, we generalize Lin-Lu-Yau's Ricci curvature to weighted graphs and give a simple limit-free definition. We prove two extremal results on the sum of Ricci curvatures for weighted graph. A weighted graph $G=(V,E,d)$ is an undirected graph $G=(V,E)$ associated with a distance function $d\colon E\to [0,\infty)$. By redefining the weights if possible, without loss of generality, we assume that the shortest weighted distance between $u$ and $v$ is exactly $d(u,v)$ for any edge $uv$. Now consider a random walk whose transitive probability from an vertex $u$ to its neighbor $v$ (a jump move along the edge $uv$) is proportional to $w_{uv}:=F(d(u,v))/d(u,v)$ for some given function $F(\bullet)$. We first generalize Lin-Lu-Yau's Ricci curvature definition to this weighted graph and give a simple limit-free representation of $κ(x, y)$ using a so called $\ast$-coupling functions. The total curvature $K(G)$ is defined to be the sum of Ricci curvatures over all edges of $G$. We proved the following theorems: if $F(\bullet)$ is a decreasing function, then $K(G)\geq 2|V| -2|E|$; if $F(\bullet)$ is an increasing function, then $K(G)\leq 2|V| -2|E|$. Both equalities hold if and only if $d$ is a constant function plus the girth is at least $6$. In particular, these imply a Gauss-Bonnet theorem for (unweighted) graphs with girth at least $6$, where the graph Ricci curvature is defined geometrically in terms of optimal transport.
We study the existence of solutions of Ricci flow equations of Ollivier-Lin-Lu-Yau curvature defined on weighted graphs. Our work is motivated by\cite{NLLG} in which the discrete time Ricci flow algorithm has been applied successfully as a discrete geometric approach in detecting complex networks. Our main result is the existence and uniqueness theorem for solutions to a continuous time normalized Ricci flow. We also display possible solutions to the Ricci flow on path graph and prove the Ricci flow on finite star graph with at least three leaves converges to constant-weighted star.
In this paper, we consider the Turán problems on $\{1,3\}$-hypergraphs. We prove that a $\{1, 3\}$-hypergraph is degenerate if and only if it's $H^{\{1, 3\}}_5$-colorable, where $H^{\{1, 3\}}_5$ is a hypergraph with vertex set $V=[5]$ and edge set $E=\{\{2\}, \{3\}, \{1, 2, 4\}, \{1, 3, 5\}, \{1, 4, 5\}\}.$ Using this result, we further prove that for any finite set $R$ of distinct positive integers, except the case $R=\{1, 2\}$, there always exist non-trivial degenerate $R$-graphs. We also compute the Turán densities of some small $\{1,3\}$-hypergraphs.
For any $r$-order $\{0, 1\}$-tensor $A$ with $e$ ones, we prove that the spectral radius of $A$ is at most $e^{\frac{r-1}{r}}$ with the equality holds if and only if $e={k^r}$ for some integer $k$ and all ones forms a principal sub-tensor ${\bf 1}_{k\times \cdots \times k}$. We also prove a stability result for general tensor $A$ with $e$ ones where $e=k^r+l$ with relatively small $l$. Using the stability result, we completely characterized the tensors achieving the maximum spectral radius among all $r$-order $\{0, 1\}$-tensor $A$ with $k^r+l$ ones, for $-r-1\leq l \leq r$, and $k$ sufficiently large.
Tanglegrams are special graphs that consist of a pair of rooted binary trees with the same number of leaves, and a perfect matching between the two leaf-sets. These objects are of use in phylogenetics and are represented with straightline drawings where the leaves of the two plane binary trees are on two parallel lines and only the matching edges can cross. The tangle crossing number of a tanglegram is the minimum crossing number over all such drawings and is related to biologically relevant quantities, such as the number of times a parasite switched hosts. Our main results for tanglegrams which parallel known theorems for crossing numbers are as follows. The removal of a single matching edge in a tanglegram with $n$ leaves decreases the tangle crossing number by at most $n-3$, and this is sharp. Additionally, if $γ(n)$ is the maximum tangle crossing number of a tanglegram with $n$ leaves, we prove $\frac{1}{2}\binom{n}{2}(1-o(1))\leγ(n)<\frac{1}{2}\binom{n}{2}$. Further, we provide an algorithm for computing non-trivial lower bounds on the tangle crossing number in $O(n^4)$ time. This lower bound may be tight, even for tanglegrams with tangle crossing number $Θ(n^2)$.
For $r\geq 3$, let $f_r\colon [0,\infty)\to [1,\infty)$ be the unique analytic function such that $f_r({k\choose r})={k-1\choose r-1}$ for any $k\geq r-1$. We prove that the spectral radius of an $r$-uniform hypergraph $H$ with $e$ edges is at most $f_r(e)$. The equality holds if and only if $e={k\choose r}$ for some positive integer $k$ and $H$ is the union of a complete $r$-uniform hypergraph $K_k^r$ and some possible isolated vertices. This result generalizes the classical Stanley's theorem on graphs.