Shiping Liu

An extremal theorem for positive curvature of graphs

Kaizhe Chen, Shiping Liu, Zhe You

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\rce…

New bounds for equiangular lines and Balla's conjecture

Chuanyuan Ge, Shiping Liu

Let $N_{α} (d)$ denote the maximum number of equiangular lines in $R^{d}$ with common angle $arccos (α)$ . Balla conjectured that, if the spectral radius order $κ_{\frac{1-α}{2α…

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

Kaizhe Chen, Shiping Liu, Zhe You

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…

On Lichnerowicz sharp distance-regular graphs

Kaizhe Chen, Shiping Liu, Heng Zhang

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 Lichnerowic…

A new proof of the Lemmens-Seidel conjecture

Chuanyuan Ge, Shiping Liu

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 re…

On finiteness of spectral radius order

Yanlong Ding, Chuanyuan Ge, Shiping Liu

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…

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

Shiping Liu, Qing Xia

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…

Edge-connectivity of graphs with non-negative Bakry-Émery curvature and amply regular graphs

Kaizhe Chen, Jack H. Koolen, Shiping Liu

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. Pre…

Equiangular lines via nodal domains

Chuanyuan Ge, Shiping Liu

For given $Δ > 0$ and $0 < λ < 3/ 2$ , 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…

Halin graphs with positive Lin-Lu-Yau curvature

Kaizhe Chen, Huiqiu Lin, Shiping Liu +1

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 c…

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

Shiping Liu, Heng Zhang

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 thi…

Connectivity versus Lin-Lu-Yau curvature

Kaizhe Chen, Shiping Liu, Zhe You

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…

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

Kaizhe Chen, Shiping Liu

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 upo…

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

Ioannis Ivrissimtzis, Carsten Lange, Shiping Liu +1

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…

Curvature and local matchings of conference graphs and extensions

Kaizhe Chen, Shiping Liu, Heng Zhang

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 $(4 γ + 1, 2 γ, γ - 1, γ)$ . Ou…

Discrete trace formulas and holomorphic functional calculus for the adjacency matrix of regular graphs

Yulin Gong, Wenbo Li, Shiping Liu

We provide a unified method to study the adjacency matrices of regular graphs (including infinite ones) using holomorphic functional calculus. By applying this calculus on a specif…

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

Yulin Gong, Wenbo Li, Shiping Liu

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 ge…

A note on Steinerberger's curvature for graphs

David Cushing, Supanat Kamtue, Erin Law +3

In this note, we provide Steinerberger curvature formulas for block graphs, discuss curvature relations between two graphs and the graph obtained by connecting them via a bridge, a…