arXiv subjects

Shiping Liu

Publications and source records attributed to Shiping Liu.

At least 19 recordsLinked to original sources

arXiv2026

A rigidity theorem for edge-connectivity under non-negative Bakry--Émery curvature

The edge-connectivity $κ(G)$ of a graph $G$ is the minimum number of edges whose removal disconnects $G$, and it is bounded above by the minimum degree $δ(G)$. We classify all connected locally finite simple graphs with nonnegative Bakry--Émery curvature satisfying $κ(G)<δ(G)$. All such graphs are infinite. Consequently, every finite connected simple graph with nonnegative Bakry--Émery curvature satisfies $κ(G)=δ(G)$, confirming a conjecture of Chen, Koolen, and Liu.

math.CO↗

arXiv2026

Volume Doubling and Geometric Inequalities on Graphs with Bakry-Émery Curvature

We consider irreducible reversible Markov chains whose nonzero off-diagonal transition probabilities are bounded below by $ε>0$ and whose reversible measure satisfies $m_-\le m(x)\leκm_-$. Under the Bakry--Émery curvature condition $\mathrm{CD}(K,\infty)$, with $K\in\mathbb{R}$, we establish Wang's log-Harnack inequality $$ P_t(\log f)(x)\le\log P_tf(y) +\frac{C\mathrm{dist}(x,y)^2}{\int_0^t e^{2Ks}\mathrm{d} s}, $$ for bounded positive $f$ bounded away from zero and all $x,y\in V$, where $C$ is a constant dependent only on $\varepsilon$. Building upon this inequality, we further prove volume doubling property: let $K_-:=\max\{-K,0\}$ $$ \mathrm{Vol}(B(o,2R))\le C_0\exp\!\bigl(C_1\sqrt{K_-}\,R\bigr) \mathrm{Vol}(B(o,R)),\qquad R>0, $$ where $C_0,C_1$ depend only on $ε,κ$. If $d$ is the maximum combinatorial degree of the support graph, we further prove under $\mathrm{CD}(0,\infty)$ the sharp volume growth bound $\mathrm{Vol}(B(o,R))\le m_-CR^{\lfloor d/2\rfloor}$ for all $o$ and $R\geq 1$.

math.PR↗

arXiv2026

Entropy estimates and volume rigidity for graphs with positive Bakry-Émery curvature

We establish a volume bound for weighted graphs with positive Bakry--Émery curvature through entropy estimates for the heat flow and characterize the equality case. As a consequence, if a connected unweighted graph $G=(V,E)$ satisfies the Bakry--Émery curvature--dimension condition $CD(2,\infty)$ with respect to the unnormalized Laplacian, then $$ |V|\le 2^d, $$ where $d$ denotes the maximum degree. Equality holds if and only if $G$ is a hypercube. We also establish an analogous volume rigidity result for the normalized Laplacian.

math.DG↗

arXiv2026

New graph invariants based on $p$-Laplacian eigenvalues

We present monotonicity inequalities for certain functions involving eigenvalues of $p$-Laplacians on signed graphs with respect to $p$. Inspired by such monotonicity, we propose new spectrum-based graph invariants, called (variational) cut-off adjacency eigenvalues, that are relevant to certain eigenvector-dependent nonlinear eigenvalue problem. Using these invariants, we obtain new lower bounds for the $p$-Laplacian variational eigenvalues, essentially giving the state-of-the-art spectral asymptotics for these eigenvalues. Moreover, based on such invariants, we establish two inertia bounds regarding the cardinalities of a maximum independent set and a minimum edge cover, respectively. The first inertia bound enhances the classical Cvetković bound, and the second one implies that the $k$-th $p$-Laplacian variational eigenvalue is of the order $2^p$ as $p$ tends to infinity whenever $k$ is larger than the cardinality of a minimum edge cover of the underlying graph. We further discover an interesting connection between graph $p$-Laplacian eigenvalues and tensor eigenvalues and discuss applications of our invariants to spectral problems of tensors.

math.SP↗

arXiv2026

A new proof of the Lemmens-Seidel conjecture

In this paper, we give a new proof of the Lemmens-Seidel conjecture on the maximum number of equiangular lines with a common angle $\arccos(1/5)$. This conjecture was previously resolved by Cao, Koolen, Lin, and Yu in 2022 through an analysis involving forbidden subgraphs for the smallest Seidel eigenvalue $-5$. Our new proof is based on bounds on eigenvalue multiplicities of graphs with degree no larger than $14$. To control the maximum degree of the graph associated with equiangular lines, we employ a recent inequality of Balla derived by matrix projection techniques. Our strategy also leads to a new proof for the classical result obtained by Lemmens and Seidel in 1973 for the case where the common angle is $\arccos(1/3)$.

math.CO↗

arXiv2026

Structure theorems for Lichnerowicz-sharp graphs

Hypercube graphs are fundamental model spaces of positive curvature in discrete comparison geometry. Let $G$ be a finite, connected, simple, unweighted graph with Bakry--Émery curvature bounded below by $K$. We call $G$ Lichnerowicz-sharp if its first non-zero non-normalized Laplacian eigenvalue $λ_1=K$. We prove that, after removing a canonical collection of edges on which every $K$-eigenfunction is constant, the resulting graph has a canonical bundle structure. Its fibers are regular, have similar structure with hypercubes, and are Laplacian-cospectral with hypercubes, although they need not themselves be hypercubes. If the base graph is nontrivial, then it satisfies $\mathrm{CD}(K,\infty)$ and has first eigenvalue strictly greater than $K$. As a consequence, if the vertex degree in $G$ is constant along each canonical fiber, then every fiber is a hypercube and $G$ is a hypercube bundle. Conversely, for every $d\geq 4$, we construct Lichnerowicz-sharp graphs with non-hypercube canonical fibers of degree $d$.

math.DG↗

arXiv2026

Regular Lichnerowicz-sharp graphs are hypercube bundles

Hypercube graphs are fundamental model spaces of positive curvature in discrete comparison geometry. Let $G$ be a finite, connected, simple, unweighted graph with Bakry--Émery curvature bounded below by $K$. We call $G$ Lichnerowicz-sharp if its first non-zero non-normalized Laplacian eigenvalue $λ_1=K$. We prove that all regular Lichnerowicz-sharp graphs are hypercube bundles with constant Bakry-Émery curvature $2$. A hypercube bundle is a graph bundle whose fiber graphs are hypercubes. As applications, we show that any $d$-regular Lichnerowicz-sharp graph with the multiplicity $m_K(G)$ of $λ_1=K$ at least $d/2$ can split off a hypercube of certain dimension. Moreover, we characterize all $d$-regular Lichnerowicz-sharp graphs with $m_{K}(G)\geq d-3$. Interestingly, our result leads to the following spectral rigidity theorem of hypercubes. For a graph $G$ with maximum degree $Δ$, if the multiplicity $m_K(G)\geq Δ-1$, then $G$ is a $Δ$-dimensional hypercube. This improves, in the unweighted setting, the multiplicity condition $m_K(G)\geq Δ$ appearing in the hypercube rigidity theorem of Liu, Münch, and Peyerimhoff. This improvement is optimal.

math.DG↗

arXiv2026

Artin algebras of small representation bound

This paper introduces a novel classification approach for representation-finite artin algebras in terms of the maximal length of their indecomposable modules of finite length, which we call the representation bound. To this end, we develop methods to compute almost split sequences and establish lower bounds for the lengths of the Auslander-Reiten translates of certain modules over artin algebras. We apply these techniques and results to explicitly classify artin algebras of representation bound $n$ for each positive integer $n \le 4.$

math.RT↗

arXiv2026

Equivalence of Lin--Lu--Yau curvature and 1/2-Ollivier curvature on weighted graphs

In this note, we prove that, on weighted graphs, the Lin--Lu--Yau curvature coincides with the $p$-Ollivier curvature up to scaling whenever the idleness parameter $p\geq 1/2$. Moreover, the threshold $1/2$ is sharp. This extends an earlier result of Bourne et al. (Ollivier--Ricci idleness functions of graphs, SIAM J. Discrete Math., 32 (2018), no. 2, 1408-1424), where combinatorial graphs were considered. This observation yields a simple proof for the global existence and uniqueness of solutions of the Lin--Lu--Yau curvature flow in Bai et al. (Ollivier Ricci-flow on weighted graphs, Amer. J. Math. 146 (2024), 1723-1747).

math.DG↗

arXiv2026

How Much MRI Preprocessing Is Enough? A Cost-Utility Study for Brain MRI Foundation Models

MRI preprocessing defines the input distribution seen by brain MRI foundation models, yet it is usually treated as routine data cleaning rather than a modeling choice. We ask how much preprocessing is worth its computational cost for self-supervised 3D MRI pretraining. Keeping the corpus, 3D ViT backbone, masking protocol, and downstream evaluations fixed, we compare a graded P0-P7 preprocessing spectrum for masked autoencoding (MAE) and joint-embedding predictive learning (JEPA) on 20,000 heterogeneous brain MRI volumes, then transfer the encoders to IDH prediction, MCI classification, brain age regression, and GLI/PED tumor segmentation. The results do not support a simple "more is better" rule. P0/P1 are numerically unstable, making P2 the lowest-cost feasible level; beyond P2, choosing the best feasible preprocessing level improves aggregate utility by only 3.4 percentage points for MAE and 1.8 percentage points for JEPA, with most paired gains statistically unresolved. Stronger preprocessing is beneficial only in selected regimes: IDH improves modestly, AGE and GLI/PED are often near or best at P2, and MCI shows the clearest empirical P7 gain. Cross-level MCI transfer further shows that much of the P7 advantage can be recovered by applying stronger preprocessing downstream, without requiring P7 throughout pretraining. These findings recast MRI preprocessing as a downstream-aware cost-utility decision rather than a default escalation pipeline. Code is available at https://github.com/PangJiangShuan/PreBrain.

cs.CV↗

arXiv2026

An extremal theorem for positive curvature of graphs

We prove an extremal theorem for positive Ollivier/Lin--Lu--Yau curvature: every graph of order $n\geq 8$ with more than \[ T(n)=\frac{n^2-3n}{2}-\left\lceil\frac{n}{2}\right\rceil+2 \] edges has positive Ollivier/Lin--Lu--Yau curvature, and this threshold is optimal. Moreover, for even $n\geq 12$, there exists a unique graph with $T(n)$ edges that has an edge with non-positive curvature. For $n=8,10$ and odd $n\geq 9$, the extremal graphs are not unique. This suggests a new class of extremal graph-theoretic problems arising from discrete curvature notions.

math.CO↗

arXiv2026

New bounds for equiangular lines and Balla's conjecture

Let $N_α(d)$ denote the maximum number of equiangular lines in $\mathbb{R}^d$ with common angle $\arccos(α)$. Balla conjectured that, if the spectral radius order $κ_{\frac{1-α}{2α}}$ of $\frac{1-α}{2α}$ is finite, then $$N_α(d)\leq \max\left\{\frac{(1-α^2)(1-2α^2)}{2α^4},\left\lfloor\frac{κ_{\frac{1-α}{2α}}(d-1)}{κ_{\frac{1-α}{2α}}-1}\right\rfloor\right\},$$ for any $d\geq 1$. The conjecture has previously been verified only for $α\in\left\{\frac{1}{3},\frac{1}{5},\frac{1}{1+2\sqrt{2}}\right\}$. In this paper, we prove that this conjecture holds for $α=\frac{1}{1+2\sqrt{3}}$ and $α=\sqrt{5}-2$. On the other hand, we show that Balla's conjecture fails for infinitely many $α$.

math.CO↗

arXiv2026

On Lichnerowicz sharp distance-regular graphs

The first non-zero Laplacian eigenvalue $λ_1$ of a finite graph is bounded below by its minimum Lin--Lu--Yau curvature $κ$. This is a discrete analogue of the classical Lichnerowicz Theorem. A graph with $λ_1=κ$ is called Lichnerowicz sharp. In this note, we give a new proof of the classification of Lichnerowicz sharp distance-regular graphs, which was first obtained by Münch and strengthens the corresponding classification by Cushing, Kamtue, Koolen, Liu, Münch, and Peyerimhoff, which required an extra spectral condition. As a key preparatory step, we provide a classification of all amply regular Terwilliger graphs with positive Lin--Lu--Yau curvature, a result that is interesting in its own right.

math.CO↗

arXiv2026

Criteria on forbidden subgraphs in the complements for positive Lin--Lu--Yau curvature

We investigate forbidden subgraph conditions in the complement of a graph that guarantee positive Lin--Lu--Yau curvature. In particular, we prove that every graph whose complement contains no $4$-cycles has positive Lin--Lu--Yau curvature, with the only exception of the $4$-vertex path. We further prove that, for any integer $t\ge2$, every graph on at least $\max\{t^2-2t+2, 8t\}$ vertices whose complement contains no $K_{2,t}$ has positive curvature. In addition, this lower bound on the number of vertices is optimal for $t\geq 10$. Finally, we construct examples showing that, in general, the forbidden subgraphs in these results cannot be replaced by cycles of length other than $4$ or by complete bipartite graphs $K_{s,t}$ with $s> 2$ and $t> 2$.

math.CO↗

arXiv2026

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

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.

math.CO↗

arXiv2026

Discrete Bakry-Émery curvature tensors and matrices of connection graphs

Liu, Münch, and Peyerimhoff introduced the notion of Bakry-Émery curvature for connection graphs as a means to derive Buser-type bounds on the eigenvalues of connection Laplacians. In this work, we present a reformulation of the Bakry-'Emery curvature at a vertex within a connection graph. Our approach expresses this curvature through the smallest eigenvalue of a set of unitarily equivalent curvature matrices. We interpret these matrices as representations of a newly defined curvature tensor, each corresponding to a different orthonormal basis of the vertex's tangent space. This framework significantly extends earlier studies by Cushing et al. and Siconolfi on curvature matrices of standard graphs. It is important to note that the Bakry-Émery curvature in connection graphs can behave very differently from that in the underlying graphs. For instance, constant functions generally fail to serve as eigenfunctions of the connection Laplacian, which poses a substantial challenge when attempting to generalize results from standard graphs to connection graphs. We address this issue by employing the Schur complement, applied twice using pseudoinverses. Additionally, we investigate the Bakry-Émery curvature in Cartesian products of connection graphs, extending and strengthening the earlier findings of Liu, Münch, and Peyerimhoff. While our results for vertices with locally balanced structures encompass previous work, we also shed light on intriguing behaviors that arise in locally unbalanced connection structures.

math.CO↗

arXiv2026

Curvature and local matchings of conference graphs and extensions

We prove a conjecture of Bonini et al. on the precise values of the Lin--Lu--Yau curvature of conference graphs, i.e., strongly regular graphs with parameters $(4γ+1,2γ,γ-1,γ)$. Our method depends only on the parameter relations and applies to broader classes of amply regular graphs. In particular, we develop a new combinatorial approach to show the existence of local perfect matchings. A key observation is that counting common neighbors leads to useful quadratic polynomials. As a corollary, we derive an interesting number-theoretic result concerning quadratic residues.

math.CO↗

arXiv2026

Beyond Conditional Computation: Retrieval-Augmented Genomic Foundation Models with Gengram

Current genomic foundation models (GFMs) rely on extensive neural computation to implicitly approximate conserved biological motifs from single-nucleotide inputs. We propose Gengram, a conditional memory module that introduces an explicit and highly efficient lookup primitive for multi-base motifs via a genomic-specific hashing scheme, establishing genomic "syntax". Integrated into the backbone of state-of-the-art GFMs, Gengram achieves substantial gains (up to 14%) across several functional genomics tasks. The module demonstrates robust architectural generalization, while further inspection of Gengram's latent space reveals the emergence of meaningful representations that align closely with fundamental biological knowledge. By establishing structured motif memory as a modeling primitive, Gengram simultaneously boosts empirical performance and mechanistic interpretability, providing a scalable and biology-aligned pathway for the next generation of GFMs. The code is available at https://github.com/zhejianglab/Genos, and the model checkpoint is available at https://huggingface.co/ZhejiangLab/Gengram.

q-bio.GN↗