Related papers: Cheeger Inequalities for Vertex Expansion and Rewe…

We prove a theorem that can be thought of as a common generalization of the Discrete Nodal Theorem and (one direction of) Cheeger's Inequality for graphs. special case of this result will assert that if the second and third eigenvalues of…

Let $\Gamma$ be a Cayley graph, or a Cayley sum graph, or a twisted Cayley graph, or a twisted Cayley sum graph, or a vertex-transitive graph. Denote the degree of $\Gamma$ by $d$, its edge Cheeger constant by $\mathfrak{h}_\Gamma$, and its…

Given a graph with a designated set of boundary vertices, we define a new notion of a Neumann Laplace operator on a graph using a reflection principle. We show that the first eigenvalue of this Neumann graph Laplacian satisfies a Cheeger…

We study the volume growth of metric balls as a function of the radius in discrete spaces, and focus on the relationship between volume growth and discrete curvature. We improve volume growth bounds under a lower bound on the so-called…

The discrete Cheeger inequality, due to Alon and Milman (J. Comb. Theory Series B 1985), is an indispensable tool for converting the combinatorial condition of graph expansion to an algebraic condition on the eigenvalues of the graph…

A basic fact in spectral graph theory is that the number of connected components in an undirected graph is equal to the multiplicity of the eigenvalue zero in the Laplacian matrix of the graph. In particular, the graph is disconnected if…

We study the complexity of approximating the vertex expansion of graphs $G = (V,E)$, defined as \[ \Phi^V := \min_{S \subset V} n \cdot \frac{|N(S)|}{|S| |V \backslash S|}. \] We give a simple polynomial-time algorithm for finding a subset…

We define a new Cheeger-like constant for graphs and we use it for proving Cheeger-like inequalities that bound the largest eigenvalue of the normalized Laplace operator.

We prove a lower bound for the $k$-th Steklov eigenvalues in terms of an isoperimetric constant called the $k$-th Cheeger-Steklov constant in three different situations: finite spaces, measurable spaces, and Riemannian manifolds. These…

We introduce the $st$-cut version the Sparsest-Cut problem, where the goal is to find a cut of minimum sparsity among those separating two distinguished vertices $s,t\in V$. Clearly, this problem is at least as hard as the usual (non-$st$)…

We prove a new generalization of the higher-order Cheeger inequality for partitioning with buffers. Consider a graph $G=(V,E)$. The buffered expansion of a set $S \subseteq V$ with a buffer $B \subseteq V \setminus S$ is the edge expansion…

Let \phi(G) be the minimum conductance of an undirected graph G, and let 0=\lambda_1 <= \lambda_2 <=... <= \lambda_n <= 2 be the eigenvalues of the normalized Laplacian matrix of G. We prove that for any graph G and any k >= 2, \phi(G) =…

We consider higher-dimensional generalizations of the normalized Laplacian and the adjacency matrix of graphs and study their eigenvalues for the Linial-Meshulam model $X^k(n,p)$ of random $k$-dimensional simplicial complexes on $n$…

The O(d) Synchronization problem consists of estimating a set of unknown orthogonal transformations O_i from noisy measurements of a subset of the pairwise ratios O_iO_j^{-1}. We formulate and prove a Cheeger-type inequality that relates a…

We use the concept of intrinsic metrics to give a new definition for an isoperimetric constant of a graph. We use this novel isoperimetric constant to prove a Cheeger-type estimate for the bottom of the spectrum which is nontrivial even if…

The second largest eigenvalue of a graph is an important algebraic parameter which is related with the expansion, connectivity and randomness properties of a graph. Expanders are highly connected sparse graphs. In coding theory, Expander…

On a compact metric graph, we consider the spectrum of the Laplacian defined with a mix of standard and Dirichlet vertex conditions. A Cheeger-type lower bound on the gap $\lambda_2 - \lambda_1$ is established, with a constant that depends…

The graph Cheeger constant and Cheeger inequalities are generalized to the case of hypergraphs whose edges have the same cardinality. In particular, it is shown that the second largest eigenvalue of the generalized normalized Laplacian is…

In this paper, we give tight bounds for the normalized Laplacian eigenvalues of hypergraphs that are not necessarily uniform, and provide an edge version interlacing theorem, a Cheeger inequality, and a discrepancy inequality that are…

The Cheeger constant of a graph is the smallest possible ratio between the size of a subgraph and the size of its boundary. It is well known that this constant must be at least $\frac{\lambda_1}{2}$, where $\lambda_1$ is the smallest…