Related papers: Sharp isoperimetric inequalities for infinite plan…
We study upper bounds for the first non-zero eigenvalue of the Steklov problem defined on finite graphs with boundary. For finite graphs with boundary included in a Cayley graph associated to a group of polynomial growth, we give an upper…
We obtain several sharp spectral bounds, approximations, and exact values for the isoperimetric number and related edge-expansion parameters of graphs. Our results focus on graph powers and on families of graphs with rich algebraic or…
A celebrated theorem of Kanai states that quasi-isometries preserve isoperimetric inequalities between uniform Riemannian manifolds (with positive injectivity radius) and graphs. Our main result states that we can study the (Cheeger)…
Let $G$ be a graph which satisfies $c^{-1} a^r \le |B(v,r)| \le c a^r$, for some constants $c,a>1$, every vertex $v$ and every radius $r$. We prove that this implies the isoperimetric inequality $|\partial A| \ge C |A| / \log(2+ |A|)$ for…
Given a plane geometric graph $G$ on $n$ vertices, we want to augment it so that given parity constraints of the vertex degrees are met. In other words, given a subset $R$ of the vertices, we are interested in a plane geometric supergraph…
We give an upper bound on the number of perfect matchings in an undirected simple graph $G$ with an even number of vertices, in terms of the degrees of all the vertices in $G$. This bound is sharp if $G$ is a union of complete bipartite…
Let G = (V,E) be a finite, simple and undirected graph. For $S \subseteq V$, let $\delta(S,G) = \{(u,v) \in E : u \in S \mbox {and} v \in V-S \}$ be the edge boundary of $S$. Given an integer $i$, $1 \leq i \leq | V |$, let the edge…
In this short note, we establish an edge-isoperimetric inequality for arbitrary product graphs. Our inequality is sharp for subsets of many different sizes in every product graph. In particular, it implies that the $2^d$-element sets with…
We prove a relative isoperimetric inequality in the plane, when the perimeter is defined with respect to a convex, positively homogeneous function of degree one, and characterize the minimizers.
The $k$-th symmetric product of a graph $G$ with vertex set $V$ with edge set $E$ is a graph with vertices as $k$-sets of $V$, where two $k$-sets are connected by an edge if and only if their symmetric difference is an edge in $E$. Using…
We obtain new sharp isoperimetric inequalities on a Riemannian manifold equipped with a probability measure, whose generalized Ricci curvature is bounded from below (possibly negatively), and generalized dimension and diameter of the convex…
We prove that for every integer $t\geq 1$, the class of intersection graphs of curves in the plane each of which crosses a fixed curve in at least one and at most $t$ points is $\chi$-bounded. This is essentially the strongest…
The geometric thickness of a graph G is the minimum integer k such that there is a straight line drawing of G with its edge set partitioned into k plane subgraphs. Eppstein [Separating thickness from geometric thickness. In: Towards a…
Let $\Phi$ be a unital positive linear map and let $A$ be a positive invertible operator. We prove that there exist partial isometries $U$ and $V$ such that \[ |\Phi(f(A))\Phi(A)\Phi(g(A))|\leq U^*\Phi(f(A)Ag(A))U \] and…
A purely analytic proof is given for an inequality that has as a direct consequence the two most important affine isoperimetric inequalities of plane convex geometry: The Blaschke-Santalo inequality and the affine isoperimetric inequality…
We introduce the Density Formula for (topological) drawings of graphs in the plane or on the sphere, which relates the number of edges, vertices, crossings, and sizes of cells in the drawing. We demonstrate its capability by providing…
This paper considers affine analogues of the isoperimetric inequality in the sense of piecewise linear topology. Given a closed polygon P embedded in R^d having n edges, we give upper and lower bounds for the minimal number of triangles…
Houdr\'e and Tetali defined a class of isoperimetric constants $\varphi_p$ of graphs for $0 \leq p \leq 1$, and conjectured a Cheeger-type inequality for $\varphi_\frac12$ of the form $$\lambda_2 \lesssim \varphi_\frac12 \lesssim…
A graph is 1-planar if it can be drawn in the plane so that each edge is crossed by at most one another edge. In this work we prove that each 1-planar graph of minimum degree at least $3$ contains an edge with degrees of its endvertices of…
An "edge guard set" of a plane graph $G$ is a subset $\Gamma$ of edges of $G$ such that each face of $G$ is incident to an endpoint of an edge in $\Gamma$. Such a set is said to guard $G$. We improve the known upper bounds on the number of…