arXiv subjects

Chunyang Hu

Publications and source records attributed to Chunyang Hu.

5 recordsLinked to original sources

arXiv2026

A brief note about p-curvature on graphs

In this paper, we consider Wang's $CD_p(m,K)$ condition on graphs, which depends on the $p$-Laplacian $\Delta_p$ for $p>1$ and is an extension of the classical Bakry-\'Emery $CD(m,K)$ curvature dimension condition. We calculate several examples including paths, cycles and star graphs, and we show that the $p$-curvature is non-negative at some vertices in the case $p\geq 2$, while it approaches to $-\infty$ in the case of $1 2$. As a consequence, an analogous proof that non-negative curvature is preserved under taking Cartesian products is not possible for $p > 2$.

math.CO↗

arXiv2025

On a magneto-spectral invariant on finite graphs

In this paper, we introduce a magneto-spectral invariant for finite graphs. This invariant vanishes on trees and is maximized by complete graphs. We compute this invariant for cycles, complete graphs, wheel graphs, hypercubes, complete bipartite graphs and suspensions of trees and derive various lower and upper bounds. In particular, we provide a sharp upper bound for regular bipartite graphs and derive a direct relation between the class of graphs assuming this upper bound and the class of unit weighing matrices, which are generalizations of complex Hadamard matrices. Moreover, this class of bipartite graphs has non-negative magnetic Bakry-\'Emery curvature and is preserved under both the Cartesian product and a partial tensor product for bipartite graphs. The study of our invariant for certain pairs of cospectral graphs indicates also that this invariant allows us to distinguish between them. Finally, we discuss the behaviour of this invariant under various graph operations and investigate relations to the spectral gap.

math.SP↗

arXiv2024

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 $(n,d,\alpha,\beta)$ satisfying $\alpha\leq 6\beta-9$. We achieve these results by a significantly improved Lin--Lu--Yau curvature estimate and new Bakry--\'Emery 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↗

arXiv2023

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, Houdr\'e 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↗

arXiv2022

Discrete Bakry-\'Emery curvature tensors and matrices of connection graphs

Liu, M\"unch, and Peyerimhoff introduced the notion of Bakry-\'Emery 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-\'Emery 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-\'Emery curvature in Cartesian products of connection graphs, extending and strengthening the earlier findings of Liu, M\"unch, 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↗