Related papers: Quantitative Obata's theorem in discrete setting

We derive a number of upper and lower bounds for the first nontrivial eigenvalue of a finite quantum graph in terms of the edge connectivity of the graph, i.e., the minimal number of edges which need to be removed to make the graph…

We study the approximation of eigenvalues for the Laplace-Beltrami operator on closed Riemannian manifolds in the class $\mathcal{M}$, characterized by bounded Ricci curvature, a lower bound on the injectivity radius, and an upper bound on…

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 hypercube. The proofs use well-known semigroup methods as…

This paper establishes quantitative high-probability bounds on the eigenvalues and eigenfunctions of $\epsilon$-neighborhood graph Laplacians constructed from i.i.d. random variables on $m$-dimensional closed Riemannian manifolds $(M,g)$…

We prove that the maximum eigenvalue of the (both signed and unsigned) Laplacian of level $k$ Kikuchi graph of any graph $G$ with $m$ edges is at most $m+k$. This confirms four recent conjectures of Apte, Parekh, and Sud. As applications,…

In this paper, we investigate complete Riemannian manifolds satisfying the lower weighted Ricci curvature bound $\mathrm{Ric}_{N} \geq K$ with $K>0$ for the negative effective dimension $N<0$. We analyze two $1$-dimensional examples of…

In this paper, we prove that any simple $\{C_3,C_5\}$-free non-empty connected graph $G$ with LLY curvature bounded below by $\kappa>0$ has the order at most $2^{\frac{2}{\kappa}}$. This upper bound is achieved if and only if $G$ is a…

In this work we show that given a connectivity graph $G$ of a $[[n,k,d]]$ quantum code, there exists $\{K_i\}_i, K_i \subset G$, such that $\sum_i |K_i|\in \Omega(k), \ |K_i| \in \Omega(d)$, and the $K_i$'s are $\tilde{\Omega}(…

We prove Obata's rigidity theorem for metric measure spaces that satisfy a Riemannian curvature-dimension condition. Additionally, we show that a lower bound $K$ for the generalized Hessian of a sufficiently regular function $u$ holds if…

Let $G$ be a graph, and let $\lambda(G)$ denote the smallest eigenvalue of $G$. First, we provide an upper bound for $\lambda(G)$ based on induced bipartite subgraphs of $G$. Consequently, we extract two other upper bounds, one relying on…

We use two variational techniques to prove upper bounds for sums of the lowest several eigenvalues of matrices associated with finite, simple, combinatorial graphs. These include estimates for the adjacency matrix of a graph and for both…

In this paper, we generalize Lin-Lu-Yau's Ricci curvature to weighted graphs and give a simple limit-free definition. We prove two extremal results on the sum of Ricci curvatures for weighted graph. A weighted graph $G=(V,E,d)$ is an…

A connected undirected graph $G = (V,E)$ is lower conformally rigid if uniform edge weights maximize the second smallest Laplacian eigenvalue $\lambda_2(w)$ over all normalized edge weights $w$, and upper conformally rigid if uniform edge…

We investigate quantum graphs with infinitely many vertices and edges without the common restriction on the geometry of the underlying metric graph that there is a positive lower bound on the lengths of its edges. Our central result is a…

We investigate static metrics on simple manifolds with compact boundary and establish an Obata-type rigidity theorem. We identify new sufficient geometric conditions under which the combined curvature map $g\mapsto (R_g, H_g)$ is a local…

We derive upper bounds for the eigenvalues of the Kirchhoff Laplacian on a compact metric graph depending on the graph's genus g. These bounds can be further improved if $g = 0$, i.e. if the metric graph is planar. Our results are based on…

For every pair of vertices u and v with d(u; v) = n, Wun G v denotes the set of all vertices of G that are closer to u than to v. In this paper, we introduce quasi-({\lambda}; n)-distance-balanced graphs and then study some properties of…

We prove that the diameter of any unweighted connected graph G is O(k log n/lambda_k), for any k>= 2. Here, lambda_k is the k smallest eigenvalue of the normalized laplacian of G. This solves a problem posed by Gil Kalai.

We show that a compact quaternionic contact manifold of dimension seven that satisfies a Lichnerowicz-type lower Ricci-type bound and has the $P$-function of any eigenfunction of the sub-Laplacian non-negative achieves its smallest possible…

This paper extends the Bakry-\'{E}mery theorem connecting the Ricci curvature and log-Sobolev inequalities to the matrix-valued setting. Using tools from noncommuative geometry, it is shown that for a right invariant second order…