121 citations · 538 across the 72 of their papers we have counts for
17 papers · 1 filter
Structure theorems for Lichnerowicz-sharp graphs
Yanlong Ding, Shiping Liu, Chiyu Zhou
Hypercube graphs are fundamental model spaces of positive curvature in discrete comparison geometry. Let be a finite, connected, simple, unweighted graph with Bakry--Émery curv…
Equivalence of Lin--Lu--Yau curvature and 1/2-Ollivier curvature on weighted graphs
Shiping Liu, Yunyan Yang
In this note, we prove that, on weighted graphs, the Lin--Lu--Yau curvature coincides with the -Ollivier curvature up to scaling whenever the idleness parameter . Mor…
Regular Lichnerowicz-sharp graphs are hypercube bundles
Yanlong Ding, Shiping Liu, Chiyu Zhou
Hypercube graphs are fundamental model spaces of positive curvature in discrete comparison geometry. Let be a finite, connected, simple, unweighted graph with Bakry--Émery curv…
A generalized Cheeger inequality and the Steklov Problem on finite graphs
Bobo Hua, Supanat Kamtue, Shiping Liu +2
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 Che…
Quantitative Obata's theorem in discrete setting
Shiping Liu, Chiyu Zhou
Under mild assumptions, we show that a connected weighted graph with lower Ricci curvature bound in the sense of Bakry-Émery and the -th non-zero Laplacian eigenvalue…
Ricci curvature, diameter and eigenvalues of amply regular graphs
Kaizhe Chen, Chunyang Hu, Shiping Liu +1
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…
Entropic curvature not comparable to other curvatures -- or is it?
Supanat Kamtue, Shiping Liu, Florentin Münch +1
In this paper we consider global -curvatures of finite Markov chains with associated means in the spirit of the entropic curvature (based on the logarithmic mean) by Erbar-M…
Bakry-Émery calculus for entropic curvature, new diameter estimates, and spectral gaps
Supanat Kamtue, Shiping Liu, Florentin Münch +1
In this paper, we propose a generalization of Bakry-Émery's calculus which allows us to formulate both Bakry-Émery and entropic curvature simultaneously. This formulation represent…
Graph bundles and Ricci-flatness
Wenbo Li, Shiping Liu
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 n…
Comparative analysis of two discretizations of Ricci curvature for complex networks
Areejit Samal, R. P. Sreejith, Jiao Gu +3
We have performed an empirical comparison of two distinct notions of discrete Ricci curvature for graphs or networks, namely, the Forman-Ricci curvature and Ollivier-Ricci curvatur…
A note on eigenvalue bounds for non-compact manifolds
Matthias Keller, Shiping Liu, Norbert Peyerimhoff
In this article we prove upper bounds for the Laplace eigenvalues below the essential spectrum for strictly negatively curved Cartan-Hadamard manifolds. Our bound is given in…
Rigidity properties of the hypercube via Bakry-Emery curvature
Shiping Liu, Florentin Münch, Norbert Peyerimhoff
We give rigidity results for the discrete Bonnet-Myers diameter bound and the Lichnerowicz eigenvalue estimate. Both inequalities are sharp if and only if the underlying graph is a…
Distance bounds for graphs with some negative Bakry-Émery curvature
Shiping Liu, Florentin Münch, Norbert Peyerimhoff +1
We prove distance bounds for graphs possessing positive Bakry-Émery curvature apart from an exceptional set, where the curvature is allowed to be non-positive. If the set of non-po…
Ricci curvature and eigenvalue estimates for the magnetic Laplacian on manifolds
Michela Egidi, Shiping Liu, Florentin Münch +1
In this paper, we present a Lichnerowicz type estimate and (higher order) Buser type estimates for the magnetic Laplacian on a closed Riemannian manifold with a magnetic potential.…
Bakry-Emery curvature and diameter bounds on graphs
Shiping Liu, Florentin Münch, Norbert Peyerimhoff
We prove diameter bounds for graphs having positive Ricci-curvature bound in Bakry-Emery sense. One result using only curvature and maximal vertex degree is sharp in case of hyperc…
An optimal dimension-free upper bound for eigenvalue ratios
Shiping Liu
On a closed weighted Riemannian manifold with nonnegative Bakry-Émery Ricci curvature, it is shown that the ratio of the -th to first eigenvalues of the weighted Laplacian is do…
Liouville theorems for -harmonic maps into Hadamard spaces
Bobo Hua, Shiping Liu, Chao Xia
In this paper, we study harmonic functions on weighted manifolds and harmonic maps from weighted manifolds into Hadamard spaces introduced by Korevaar and Schoen. We prove Liouvill…