Related papers: A Cheeger inequality for the lower spectral gap

In this paper, we study distance-regular graphs $\Gamma$ that have a pair of distinct vertices, say x and y, such that the number of common neighbors of x and y is about half the valency of $\Gamma$. We show that if the diameter is at least…

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 present a finer quantitative version of an observation due to Breuillard, Green, Guralnick and Tao which tells that for finite non-bipartite Cayley graphs, once the nontrivial eigenvalues of their normalized adjacency matrices are…

J.H. Koolen and J. Park have proved a lower bound for intersection number $c_2$ of a distance-regular graph $\Gamma$. Moreover, they showed that the graph $\Gamma$ which attains the equality in this bound is a Terwilliger graph. We prove…

Given a finite transitive permutation group $G\leq \operatorname{Sym}(\Omega)$, with $|\Omega|\geq 2$, the derangement graph $\Gamma_G$ of $G$ is the Cayley graph $\operatorname{Cay}(G,\operatorname{Der}(G))$, where $\operatorname{Der}(G)$…

We introduce a set of multi-way dual Cheeger constants and prove universal higher-order dual Cheeger inequalities for eigenvalues of normalized Laplace operators on weighted finite graphs. Our proof proposes a new spectral clustering…

The $\gamma$-graph of a graph $G$ is the graph whose vertices are labelled by the minimum dominating sets of $G$, in which two vertices are adjacent when their corresponding minimum dominating sets (each of size $\gamma(G)$) intersect in a…

We prove Cheeger inequalities for p-Laplacians on finite and infinite weighted graphs. Unlike in previous works, we do not impose boundedness of the vertex degree, nor do we restrict ourselves to the normalized Laplacian and, more…

We introduce uniformly vertex-transitive graphs as vertex-transitive graphs satisfying a stronger condition on their automorphism groups, motivated by a problem which arises from a Sinkhorn-type algorithm. We use the derangement graph…

For a graph $G=(V,E)$, let $bc(G)$ denote the minimum number of pairwise edge disjoint complete bipartite subgraphs of $G$ so that each edge of $G$ belongs to exactly one of them. It is easy to see that for every graph $G$, $bc(G) \leq n…

The bipartite-hole-number of a graph $G$, denoted as $\widetilde{\alpha}(G)$, is the minimum number $k$ such that there exist positive integers $s$ and $t$ with $s+t=k+1$ with the property that for any two disjoint sets $A,B\subseteq V(G)$…

The celebrated Cheeger's Inequality establishes a bound on the edge expansion of a graph via its spectrum. This inequality is central to a rich spectral theory of graphs, based on studying the eigenvalues and eigenvectors of the adjacency…

In this paper, we use machine learning to show that the Cheeger constant of a connected regular graph has a predominant linear dependence on the largest two eigenvalues of the graph spectrum. We also show that a trained deep neural network…

In this paper we study spectral properties of graphs which are constructed from two given invertible graphs by bridging them over a bipartite graph. We analyze the so-called HOMO-LUMO spectral gap which is the difference between the…

The splitting field $\mathbb{SF}(\Gamma)$ of a mixed graph $\Gamma$ is the smallest field extension of $\mathbb{Q}$ which contains all eigenvalues of the Hermitian adjacency matrix of $\Gamma$. The extension degree…

Let M be a bounded domain of a Euclidian space with smooth boundary. We relate the Cheeger constant of M and the conductance of a neighborhood graph defined on a random sample from M. By restricting the minimization defining the latter over…

For a graph $G=(V,E)$, let $\tau(G)$ denote the minimum number of pairwise edge disjoint complete bipartite subgraphs of $G$ so that each edge of $G$ belongs to exactly one of them. It is easy to see that for every graph $G$, $\tau(G) \leq…

We develop the notion of higher Cheeger constants for a measurable set $\Omega \subset \mathbb{R}^N$. By the $k$-th Cheeger constant we mean the value \[h_k(\Omega) = \inf \max \{h_1(E_1), \dots, h_1(E_k)\},\] where the infimum is taken…

We prove a $\Gamma$-convergence result for Cheeger energies along sequences of metric measure spaces, where the measure space is kept fixed, while distances are monotonically converging from below to the limit one. As a consequence, we show…

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…