arXiv subjects

Shiping Liu

Publications and source records attributed to Shiping Liu.

At least 37 records · Page 2Linked to original sources

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

Diameter bounds for distance-regular graphs via long-scale Ollivier Ricci curvature

In this paper, we derive new sharp diameter bounds for distance regular graphs, which better answer a problem raised by Neumaier and Penji\' c in many cases. Our proof is built upon a relation between the diameter and long-scale Ollivier Ricci curvature of a graph, which can be considered as an improvement of the discrete Bonnet-Myers theorem. Our method further leads to significant improvements of existing diameter bounds for amply regular graphs and $(s,c,a,k)$-graphs.

math.CO↗

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-Émery and the $d$-th non-zero Laplacian eigenvalue $λ_d$ 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

Monotonic normalized heat diffusion for distance-regular graphs with classical parameters of diameter $3$

We prove the monotonic normalized heat diffusion property on distance-regular graphs with classical parameters of diameter $3$. Regev and Shinkar found a Cayley graph for which this property fails. On the other hand, this property has been proved on abelian Cayley graphs, graphs with $3$ distinct eigenvalues and regular bipartite graphs with $4$ distinct eigenvalues by Price, Nica and Kubo-Namba, respectively. A distance regular graph with classical parameters of diameter $3$ has $4$ distinct eigenvalues and is not necessarily bipartite or vertex transitive.

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-Émery curvature and amply regular graphs

We establish a sharp edge-connectivity estimate for graphs with non-negative Bakry-Émery 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-Émery 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 $Δ>0$ and $0<λ<3/\sqrt{2}$, we show that the maximum multiplicity that $λ$ can appear as the second largest eigenvalue of a connected graph with maximum degree at most $Δ$ is $O_{Δ,λ}(1)$. 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 $0<λ<3/\sqrt{2}$, 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↗

arXiv2025

Ricci curvature, diameter and eigenvalues of amply regular graphs

Amply regular graphs are graphs with local distance-regularity constraints. In this paper, we prove a weaker version of a conjecture proposed by Qiao, Park, and Koolen on diameter bounds of amply regular graphs and make new progress on Terwilliger's conjecture on finiteness of amply regular graphs. Terwilliger's conjecture can be considered as a natural extension of the Bannai-Ito conjecture about distance-regular graphs confirmed by Bang, Dubickas, Koolen, and Moulton. As a consequence, we show that there are only finitely many amply regular graphs with parameters $(n,d,α,β)$ satisfying $α\leq 6β-9$. We achieve these results by a significantly improved Lin--Lu--Yau curvature estimate and new Bakry--Émery curvature estimates. We further discuss applications of our curvature estimates to bounding eigenvalues, isoperimetric constants, and expansion properties. In addition, we obtain a volume estimate, which is sharp for hypercubes.

math.DG↗

arXiv2025

Representation theory of hereditary artin algebras of finite representation type

Let $H$ be a hereditary artin algebra of finite representation type. We first determine all hammocks in the Auslander-Reiten quiver $\GaH$ of $\mmod H$, the category of finitely generated left $H$-modules. This enables us to obtain an effective method to construct $\GaH$ by simply viewing the ext-quiver of $H$. As easy applications, we compute the numbers of non-isomorphic indecomposable objects in $\mmod H$ and the associated cluster category $\mathscr{C}_H$, as well as the nilpotencies of the radicals of $\mmod H\hspace{-.4pt},$ $\hspace{-.5pt} D^{\hspace{.5pt}b\hspace{-.6pt}}(\hspace{-.5pt}\mmod H\hspace{-.5pt})$ and $\mathscr{C}_H$.

math.RT↗

arXiv2025

Vertex isoperimetry on signed graphs and spectra of non-bipartite Cayley and Cayley sum graphs

For a non-bipartite finite Cayley graph, we show the non-trivial eigenvalues of its normalized adjacency matrix lie in the interval $$\left[-1+\frac{ch_{out}^2}{d},1-\frac{Ch_{out}^2}{d}\right],$$ for some absolute constant $c$ and $C$, where $h_{out}$ stands for the outer vertex boundary isoperimetric constant. This improves upon recent obtained estimates aiming at a quantitative version of a result due to Breuillard, Green, Guralnick and Tao. We achieve this by extending the work of Bobkov, Houdré and Tetali on vertex isoperimetry to the setting of signed graphs. We further extend our interval estimate to the settings of vertex transitive graphs and Cayley sum graphs. As a byproduct, we answer positively open questions proposed recently by Moorman, Ralli and Tetali.

math.CO↗

arXiv2025

Halin graphs with positive Lin-Lu-Yau curvature

Halin graphs constitute an interesting class of planar and polyhedral graphs. A generalized Halin graph is obtained by connecting all leaves of a planar embedding of a tree via a cycle. A Halin graph is a generalized Halin graph having no vertex of degree two. We classify all generalized Halin graphs with positive Lin-Lu-Yau curvature.

math.CO↗

arXiv2025

Connectivity versus Lin-Lu-Yau curvature

We explore the interaction between connectivity and Lin-Lu-Yau curvature of graphs systematically. The intuition is that connected graphs with large Lin-Lu-Yau curvature also have large connectivity, and vice versa. We prove that the connectivity of a connected graph is lower bounded by the product of its minimum degree and its Lin-Lu-Yau curvature. On the other hand, if the connectivity of a graph $G$ on $n$ vertices is at least $\frac{n-1}{2}$, then $G$ has positive Lin-Lu-Yau curvature. Moreover, the bound $\frac{n-1}{2}$ here is optimal. Furthermore, we prove that the edge-connectivity is equal to the minimum vertex degree for any connected graph with positive Lin-Lu-Yau curvature. As applications, we estimate or determine the connectivity and edge-connectivity of an amply regular graph with parameters $(d,α,β)$ such that $1\neq β\geq α$.

math.CO↗

arXiv2025

Entropic curvature not comparable to other curvatures -- or is it?

In this paper we consider global $θ$-curvatures of finite Markov chains with associated means $θ$ in the spirit of the entropic curvature (based on the logarithmic mean) by Erbar-Maas and Mielke. As in the case of Bakry-Émery curvature, we also allow for a finite dimension parameter by making use of an adapted $Γ$ calculus for $θ$-curvatures. We prove explicit positive lower curvature bounds (both finite- and infinite-dimensional) for finite abelian Cayley graphs. In the case of cycles, we provide also an upper curvature bound which shows that our lower bounds are asymptotically sharp (up to a logarithmic factor). Moreover, we prove new universal lower curvature bounds for finite Markov chains as well as curvature perturbation results (allowing, in particular, to compare entropic and Bakry-Émery curvatures). Finally, we present examples where entropic curvature differs significantly from other curvature notions like Bakry-Émery curvature or Ollivier Ricci and sectional curvatures.

math.DG↗

arXiv2025

Graph bundles and Ricci-flatness

We develop a systematical way of constructing S-Ricci flat graphs which are not Abelian Cayley via graph bundle with explicit examples. For this purpose, we prove that, with some natural constrains, a non-trivial graph bundle can not be isomorphic (as graphs) to the product of the base graph and fiber graph. It stands in clear contrast to the continuous case.

math.DG↗

arXiv2024

A Representation theoretic perspective of Koszul theory

We discover a new connection between Koszul theory and representation theory. Let $\La$ be a quadratic algebra defined by a locally finite quiver with relations. Firstly, we give a combinatorial description of the local Koszul complexes and the quadratic dual $\La^!$, which enables us to describe the linear projective resolutions and the colinear injective coresolutions of graded simple $\La$-modules in terms of $\La^!$. As applications, we obtain a new class of Koszul algebras and a stronger version of the Extension Conjecture for finite dimensional Koszul algebras with a noetherian Koszul dual. Then we construct two Koszul functors, which induce a $2$-real-parameter family of pairs of derived Koszul functors between categories derived from graded $\La$-modules and those derived from graded $\La^!$-modules. In case $\La$ is Koszul, each pair of derived Koszul functors are mutually quasi-inverse, one of the pairs is Beilinson, Ginzburg and Soergel's Koszul duality. If $\La$ and $\La^!$ are locally bounded on opposite sides, then the Koszul functors induce two equivalences of bounded derived categories: one for finitely piece-supported graded modules, and one for finite dimensional graded modules. And if $\La$ and $\La^!$ are both locally bounded, then the bounded derived category of finite dimensional graded $\La$-modules has almost split triangles with the Auslander-Reiten translations and the Serre functors given by composites of derived Koszul functors.

math.RT↗

arXiv2024

Spectral convergence of random regular graphs: Chebyshev polynomials, non-backtracking walks, and unitary-color extensions

In this paper, we give a short proof of the weak convergence to the Kesten-McKay distribution for the normalized spectral measures of random $N$-lifts. This result is derived by generalizing a formula of Friedman involving Chebyshev polynomials and non-backtracking walks. We also extend a criterion of Sodin on the convergence of graph spectral measures to regular graphs of growing degree. As a result, we show that for a sequence of random $(q_n+1)$-regular graphs $G_n$ with $n$ vertices, if $q_n = n^{o(1)}$ and $q_n$ tends to infinity, the normalized spectral measure converges almost surely in $p$-Wasserstein distance to the semicircle distribution for any $p \in [1, \infty)$. This strengthens a result of Dumitriu and Pal. Many of the results are extended to unitary-colored regular graphs.

math.CO↗

arXiv2024

Polyhedral volume ratios, Izmestiev's Colin de Verdiere matrices and Spectral Gaps

We present a relation between volumes of certain lower dimensional simplices associated to a full-dimensional primal and polar dual polytope in R^k. We then discuss an application of this relation to a geometric construction of a Colin de Verdiere matrix by Ivan Izmestiev. In the second part of the paper, we introduce a variation of vertex transitive polytopes, translate their associated Colin de Verdiere matrices into random walk matrices, and investigate extremality properties of the spectral gaps of these random walk matrices in two concrete examples - permutahedra of Coxeter groups and polytopes associated to the pure rotational tetrahedral group - where maximal spectral gaps correspond to equilateral polytopes.

math.CO↗

arXiv2024

Representation theory of graded algebras given by locally finite quivers

This paper aims to study graded modules over a graded algebra $\La$ given by a locally finite quiver with homogeneous relations. By constructing a graded Nakayama functor, we discover a novel approach to establish Auslander-Reiten formulas, from which we derive almost split sequences in the category of all graded $\La$-modules. In case $\La$ is locally left (respectively, right) bounded, the category of finitely presented graded modules and that of finitely copresented graded modules both have almost split sequences on the left (respectively, right). We shall also obtain existence theorems for almost split triangles in various derived categories of graded $\La$-modules. In case $\La$ is locally bounded, an indecomposable complex in the bounded derived category of finite dimensional graded modules is the starting (respectively, ending) term of an almost split triangle if and only if it has a finite graded projective reso\-lution (respectively, injective coresolution); and consequently, this bounded derived category has almost split triangles on the right (respectively, left) if and only if every graded simple module is of finite graded projective (respectively, injective) dimension. Finally, we specialize to the existence of almost split sequences and almost split triangles for graded representations of any locally finite quiver.

math.RT↗