arXiv subjects

Shiping Liu

Publications and source records attributed to Shiping Liu.

At least 19 recordsLinked to original sources

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--\'Emery curvature bounded below by $K$. We call $G$ Lichnerowicz-sharp if its first non-zero non-normalized Laplacian eigenvalue $\lambda_1=K$. 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

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

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

math.CO↗

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--\'Emery curvature bounded below by $K$. We call $G$ Lichnerowicz-sharp if its first non-zero non-normalized Laplacian eigenvalue $\lambda_1=K$. We prove that all regular Lichnerowicz-sharp graphs are hypercube bundles with constant Bakry-\'Emery 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 $\lambda_1=K$ 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 $\Delta$, if the multiplicity $m_K(G)\geq \Delta-1$, then $G$ is a $\Delta$-dimensional hypercube. This improves, in the unweighted setting, the multiplicity condition $m_K(G)\geq \Delta$ appearing in the hypercube rigidity theorem of Liu, M\"unch, and Peyerimhoff. This improvement is optimal.

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

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

On Lichnerowicz sharp distance-regular graphs

The first non-zero Laplacian eigenvalue $\lambda_1$ of a finite graph is bounded below by its minimum Lin--Lu--Yau curvature $\kappa$. This is a discrete analogue of the classical Lichnerowicz Theorem. A graph with $\lambda_1=\kappa$ 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 M\"unch and strengthens the corresponding classification by Cushing, Kamtue, Koolen, Liu, M\"unch, 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

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↗

arXiv2025

Auslander-Reiten theory via Nakayama duality in abelian categories

Using the Nakayama duality induced by a Nakayama functor, we provide a novel and concise account of the existence of Auslander-Reiten dualities and almost split sequences in abelian categories with enough projective objects or enough injective objects. As an example, we establish the existence of almost split sequences ending with finitely presented modules and those starting with finitely copresented modules in the category of all modules over a small endo-local Hom-reflexive category. Specializing to algebras given by (not necessarily finite) quivers with relations, we further investigate when the categories of finitely presented modules, finitely copresented modules and finite dimensional modules have almost split sequences on either or both sides.

math.RT↗

arXiv2025

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↗

arXiv2025

A generalized Cheeger inequality and the Steklov Problem on finite graphs

We prove generalized Cheeger inequalities for eigenvalues of Laplacians for reversible Markov chains. Then we apply Hassannezhad and Miclo's convergence result to obtain Jammes Cheeger inequalities for Steklov eigenvalues. In particular, we get a sharp estimate for the first non-trivial Steklov eigenvalue via Escobar Cheeger constant. At the end, we extend Hassannezhad and Miclo's convergence result to non-reversible Markov chains via a different method based on resolvent convergence, answering one of their questions.

math.DG↗

arXiv2025

Quantitative Obata's theorem in discrete setting

Under mild assumptions, we show that a connected weighted graph $G$ with lower Ricci curvature bound $K>0$ in the sense of Bakry-\'Emery and the $d$-th non-zero Laplacian eigenvalue $\lambda_d$ close to $K$, with $d$ being the maximal combinatorial vertex degree of $G$, has an underlying combinatorial structure of the $d$-dimensional hypercube graph. Moreover, such a graph $G$ is close in terms of Frobenius distance to a properly weighted hypercube graph. Furthermore, we establish their closeness in terms of eigenfunctions. Our results can be viewed as discrete analogies of the almost rigidity theorem and quantitative Obata's theorem on Rimennian manifolds.

math.DG↗

arXiv2025

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↗

arXiv2025

On finiteness of spectral radius order

The concept of spectral radius order plays an crucial role in the breakthrough work on equiangular lines due to Jiang, Tidor, Yao, Zhang, and Zhao [Ann. of Math. (2) 194 (2021), no. 3, 729-743]. However, it is difficult to calculate the spectral radius order explicitly in general, or even to characterize numbers with finite spectral radius order. In this paper, we characterize numbers with finite spectral radius orders in two special classes: quadratic algebraic integers and the numbers no larger than 2. Additionally, we derive precise values of the spectral radius order of two infinite families of quadratic algebraic integers.

math.CO↗

arXiv2025

Edge-connectivity of graphs with non-negative Bakry-\'Emery curvature and amply regular graphs

We establish a sharp edge-connectivity estimate for graphs with non-negative Bakry-\'Emery curvature. This leads to a geometric criterion for the existence of a perfect matching. Precisely, we show that any regular graph with non-negative Bakry-\'Emery curvature and an even or infinite number of vertices has a perfect matching. Through a synthesis of combinatorial and curvature-related techniques, we determine the edge-connectivity of (possibly infinite) amply regular graphs.

math.CO↗

arXiv2025

Equiangular lines via nodal domains

For given $\Delta>0$ and $0<\lambda<3/\sqrt{2}$, we show that the maximum multiplicity that $\lambda$ can appear as the second largest eigenvalue of a connected graph with maximum degree at most $\Delta$ is $O_{\Delta,\lambda}(1)$. This result answers a question due to Jiang, Tidor, Yao, Zhang and Zhao [Question 6.4, Ann. of Math. (2) 194 (2021), no. 3, 729-743] in the case of $0<\lambda<3/\sqrt{2}$, and consequently leads to improvements in their results on equiangular lines. Our proof is based on the concept of nodal domains of eigenfunctions. Indeed, we establish a multiplicity estimate in terms of maximum degree and cyclomatic number of the graph, via a novel construction of eigenfunctions with large number of nodal domains.

math.CO↗