In \cite{Elek} we proved that the limit of a weakly convergent sequence of finite graphs can be viewed as a graphing or a continuous field of infinite graphs. Thus one can associate a type $II_1$-von Neumann algebra to such graph sequences. We show that in this case the integrated density of states exists that is the weak limit of the spectra of the graph Laplacians of the finite graphs is the KNS-spectral measure of the graph Laplacian of the limit graphing. Using this limit technique we prove a Cheeger type inequality for finite graphs.
We extend the definition of boundary expansion to CW complexes and prove a Cheeger-Buser-type relation between the spectral gap of the Laplacian and the boundary expansion of an orientable CW complex.
The expansion of a graph is typically associated with its spectral properties - testing whether a graph is an expander is usually done using Cheeger's inequality. One can also use multiple eigenvalues in a higher-order Cheeger's inequality to test a deeper set of properties on the graph. However Cheeger's inquality, and the higher-order Cheeger's inequality, can be imprecise tools. Recently Lee, Gharan, and Trevisan constructed the Noisy Hypercube to prove the sharpness of the gap between spectral expansion and edge expansion in the higher-order Cheeger's inequality. We are concerned with the …
In this paper, we consider a variation on Cheeger numbers related to the coboundary expanders recently defined by Dotterer and Kahle. A Cheeger-type inequality is proved, which is similar to a result on graphs due to Fan Chung. This inequality is then used to study the relationship between coboundary expanders on simplicial complexes and their corresponding eigenvalues, complementing and extending results found by Gundert and Wagner. In particular, we find these coboundary expanders do not satisfy natural Buser or Cheeger inequalities.
The Cheeger constant, $h_G$, is a measure of expansion within a graph. The classical Cheeger Inequality states: $λ_{1}/2 \le h_G \le \sqrt{2 λ_{1}}$ where $λ_1$ is the first nontrivial eigenvalue of the normalized Laplacian matrix. Hence, $h_G$ is tightly controlled by $λ_1$ to within a quadratic factor. We give an alternative Cheeger Inequality where we consider the $\infty$-norm of the corresponding eigenvector in addition to $λ_1$. This inequality controls $h_G$ to within a linear factor of $λ_1$ thereby providing an improvement to the previous quadratic bounds. An additional advantage of o…
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 inequality.
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 measure of how well it is possible to solve the O(d) synchronization problem with the spectra of an operator, the graph Connection Laplacian. We also show how this inequality provides a worst case performance guarantee for a spectral method to solve this problem.
Given an undirected graph $G$, the classical Cheeger constant, $h_G$, measures the optimal partition of the vertices into 2 parts with relatively few edges between them based upon the sizes of the parts. The well-known Cheeger's inequality states that $2 λ_1 \le h_G \le \sqrt {2 λ_1}$ where $λ_1$ is the minimum nontrivial eigenvalue of the normalized Laplacian matrix. Recent work has generalized the concept of the Cheeger constant when partitioning the vertices of a graph into $k > 2$ parts. While there are several approaches, recent results have shown these higher-order Cheeger constants to b…
Let $G$ be a finite group and $S$ be a symmetric generating set of $G$ with $|S| = d$. We show that if the undirected Cayley sum graph $C_Σ(G,S)$ is an expander graph and is non-bipartite, then the spectrum of its normalised adjacency operator is bounded away from $-1$. We also establish an explicit lower bound for the spectrum of these graphs, namely, the non-trivial eigenvalues of the normalised adjacency operator lies in the interval $\left(-1+\frac{h(G)^{4}}η, 1-\frac{h(G)^{2}}{2d^{2}}\right]$, where $h(G)$ denotes the (vertex) Cheeger constant of the $d$-regular graph $C_Σ(G,S)$ and $η= 2…
Let ϕ(G) be the minimum conductance of an undirected graph G, and let 0=λ_1 <= λ_2 <=... <= λ_n <= 2 be the eigenvalues of the normalized Laplacian matrix of G. We prove that for any graph G and any k >= 2, ϕ(G) = O(k) λ_2 / \sqrt{λ_k}, and this performance guarantee is achieved by the spectral partitioning algorithm. This improves Cheeger's inequality, and the bound is optimal up to a constant factor for any k. Our result shows that the spectral partitioning algorithm is a constant factor approximation algorithm for finding a sparse cut if λ_k$ is a constant for some constant k. This provides…
For graphs there exists a strong connection between spectral and combinatorial expansion properties. This is expressed, e.g., by the discrete Cheeger inequality, the lower bound of which states that $λ(G) \leq h(G)$, where $λ(G)$ is the second smallest eigenvalue of the Laplacian of a graph $G$ and $h(G)$ is the Cheeger constant measuring the edge expansion of $G$. We are interested in generalizations of expansion properties to finite simplicial complexes of higher dimension (or uniform hypergraphs). Whereas higher dimensional Laplacians were introduced already in 1945 by Eckmann, the generali…
We derive Cheeger inequalities for directed graphs and hypergraphs using the reweighted eigenvalue approach that was recently developed for vertex expansion in undirected graphs [OZ22,KLT22,JPV22]. The goal is to develop a new spectral theory for directed graphs and an alternative spectral theory for hypergraphs. The first main result is a Cheeger inequality relating the vertex expansion $\vecψ(G)$ of a directed graph $G$ to the vertex-capacitated maximum reweighted second eigenvalue $\vecλ_2^{v*}$: \[ \vecλ_2^{v*} \lesssim \vecψ(G) \lesssim \sqrt{\vecλ_2^{v*} \cdot \log (Δ/\vecλ_2^{v*})}. \] …
The classical Cheeger's inequality relates the edge conductance $ϕ$ of a graph and the second smallest eigenvalue $λ_2$ of the Laplacian matrix. Recently, Olesker-Taylor and Zanetti discovered a Cheeger-type inequality $ψ^2 / \log |V| \lesssim λ_2^* \lesssim ψ$ connecting the vertex expansion $ψ$ of a graph $G=(V,E)$ and the maximum reweighted second smallest eigenvalue $λ_2^*$ of the Laplacian matrix. In this work, we first improve their result to $ψ^2 / \log d \lesssim λ_2^* \lesssim ψ$ where $d$ is the maximum degree in $G$, which is optimal assuming the small-set expansion conjecture. Also…
Let $G_{\infty}=(C_m^d)_{\infty}$ denote the graph whose set of vertices is $\{1,..., m\}^d$, where two distinct vertices are adjacent iff they are either equal or adjacent in $C_m$ in each coordinate. Let $G_{1}=(C_m^d)_1$ denote the graph on the same set of vertices in which two vertices are adjacent iff they are adjacent in one coordinate in $C_m$ and equal in all others. Both graphs can be viewed as graphs of the $d$-dimensional torus. We prove that one can delete $O(\sqrt d m^{d-1})$ vertices of $G_1$ so that no topologically nontrivial cycles remain. This improves an $O(d^{\log_2 (3/2)}m…
For magnetic graphs satisfying connection curvature dimension inequality $CD^σ(n,κ)$, we prove a Harnack-type inequality for eigenfunctions of the graph magnetic Laplace operator in the manner of work done by Chung, Lin, Yau in 2014. Then we look at two applications; first a lower bound for the least eigenvalue in terms of curvature and extremal path/degree quantities, then to the magnetic Cheeger number of the graph.
In this paper we study spectra of Laplacians of infinite weighted graphs. Instead of the assumption of local finiteness we impose the condition of summability of the weight function. Such graphs correspond to reversible Markov chains with countable state spaces. We adopt the concept of the Cheeger constant to this setting and prove an analogue of the Cheeger inequality characterising the spectral gap. We also analyse the concept of the dual Cheeger constant originally introduced in \cite{B14}, which allows estimating the top of the spectrum. In this paper we also introduce a new combinatorial …
We establish mild conditions under which a possibly irregular, sparse graph $G$ has "many" strong orientations. Given a graph $G$ on $n$ vertices, orient each edge in either direction with probability $1/2$ independently. We show that if $G$ satisfies a minimum degree condition of $(1+c_1)\log_2{n}$ and has Cheeger constant at least $c_2\frac{\log_2\log_2{n}}{\log_2{n}}$, then the resulting randomly oriented directed graph is strongly connected with high probability. This Cheeger constant bound can be replaced by an analogous spectral condition via the Cheeger inequality. Additionally, we prov…
Cheeger inequalities bound the spectral gap $γ$ of a space by isoperimetric properties of that space and vice versa. In this paper, I derive Cheeger-type inequalities for nonpositive matrices (aka stoquastic Hamiltonians), real matrices, and Hermitian matrices. For matrices written $H = L+W$, where $L$ is either a combinatorial or normalized graph Laplacian, I show that: (1) when $W$ is diagonal and $L$ has maximum degree $d_{\max}$, $2h \geq γ\geq \sqrt{h^2 + d_{\max}^2}-d_\max$; (2) when $W$ is real, we can often route negative-weighted edges along positive-weighted edges such that the Cheeg…
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 Ollivier curvature, and discuss similar results under other types of discrete Ricci curvature. Following recent work in the continuous setting of Riemannian manifolds (by the first author), we then bound the eigenvalues of the Laplacian of a graph under bounds on the volume growth. In particular, the spectral gap of the graph can be bounded using a weighted disc…
We introduce the signless 1-Laplacians and the dual Cheeger constants on simplicial complexes. The connection of its spectrum to the combinatorial properties like independence number, chromatic number and dual Cheeger constant is investigated. Our estimates can be comparable to Hoffman's bounds in virtue of Laplacian on simplicial complexes. An interesting inequality relating multiplicity of the largest eigenvalue, independence number and chromatic number are provided, which could be regarded as a variant version of Lovasz sandwich theorem. Also, the behavior of the operator under the topologi…
We prove the Li-Yau gradient estimate for the heat kernel on graphs. The only assumption is a variant of the curvature-dimension inequality, which is purely local, and can be considered as a new notion of curvature for graphs. We compute this curvature for lattices and trees and conclude that it behaves more naturally than the already existing notions of curvature. Moreover, we show that if a graph has non-negative curvature then it has polynomial volume growth. We also derive Harnack inequalities and heat kernel bounds from the gradient estimate, and show how it can be used to strengthen the …
By measured graphs we mean graphs endowed with a measure on the set of vertices. In this context, we explore the relations between the appropriate Cheeger constant and Poincaré inequalities. We prove that the so-called Cheeger inequality holds in two cases: when the measure comes from a random walk, or when the measure has a bounded measure ratio. Moreover, we also prove that our measured (asymptotic) expanders are generalised expanders introduced by Tessera. Finally, we present some examples to demonstrate relations and differences between classical expander graphs and the measured ones. The …
We study the spectrum of the normalized Laplace operator of a connected graph $Γ$. As is well known, the smallest nontrivial eigenvalue measures how difficult it is to decompose $Γ$ into two large pieces, whereas the largest eigenvalue controls how close $Γ$ is to being bipartite. The smallest eigenvalue can be controlled by the Cheeger constant, and we establish a dual construction that controls the largest eigenvalue. Moreover, we find that the neighborhood graphs $Γ[l]$ of order $l\geq2$ encode important spectral information about $Γ$ itself which we systematically explore. In particular, t…
We study a natural discrete Bochner-type inequality on graphs, and explore its merit as a notion of curvature in discrete spaces. An appealing feature of this discrete version seems to be that it is fairly straightforward to compute this notion of curvature parameter for several specific graphs of interest - particularly, abelian groups, slices of the hypercube, and the symmetric group under various sets of generators. We further develop this notion by deriving Buser-type inequalities (a la Ledoux), relating functional and isoperimetric constants associated with a graph. Our derivations provid…
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 $n$ vertices. We show that for $p=Ω(\log n/n)$, the eigenvalues of these matrices are a.a.s. concentrated around two values. The main tool, which goes back to the work of Garland, are arguments that relate the eigenvalues of these matrices to those of graphs that arise as links of $(k-2)$-dimensional faces. Garland's result concerns the Laplacian; we develop an analogo…
The aim of the present article is to give an overview of spectral theory on metric graphs guided by spectral geometry on discrete graphs and manifolds. We present the basic concept of metric graphs and natural Laplacians acting on it and explicitly allow infinite graphs. Motivated by the general form of a Laplacian on a metric graph, we define a new type of combinatorial Laplacian. With this generalised discrete Laplacian, it is possible to relate the spectral theory on discrete and metric graphs. Moreover, we describe a connection of metric graphs with manifolds. Finally, we comment on Cheege…
