Related papers: Strong Isoperimetric Inequality for Tessellating Q…
This paper is devoted to the study of directed graphs with extremal properties relative to certain metric functionals. We characterize up to isomorphism critical digraphs with infinite values of diameter, quasi-diameter, radius and…
We study bond percolation for a family of infinite hyperbolic graphs. We relate percolation to the appearance of homology in finite versions of these graphs. As a consequence, we derive an upper bound on the critical probabilities of the…
The aim of this paper is to give not only an explicit upper bound of the total Q-curvature but also an induced isoperimetric deficit formula for the complete conformal metrics on $\mathbb R^n$, $n\ge 3$ with scalar curvature being…
Recently, it has been shown that, when the dimension of a graph turns out to be infinite dimensional in a broad sense, the upper critical surface and the corresponding critical behavior of an arbitrary Ising spin glass model defined over…
In this paper, replacing `equality' by 'equality almost everywhere' we modify several terms associated with the ring of measurable functions defined on a measure space $(X, \mathcal{A}, \mu)$ and thereby study the graph theoretic features…
In this paper we study the isoperimetric-type equalities for rosettes, i.e. regular closed planar curves with non-vanishing curvature. We find the exact relations between the length and the oriented area of rosettes based on the oriented…
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.
We classify instances of quantum pseudo-telepathy in the graph isomorphism game, exploiting the recently discovered connection between quantum information and the theory of quantum automorphism groups. Specifically, we show that graphs…
The equator of a graph is the length of a longest isometric cycle. We bound the order $n$ of a graph from below by its equator $q$, girth $g$ and minimum degree $\delta$ - and show that this bound is sharp when there exists a Moore graph…
In this paper, we study the asymptotic behavior of the volume of spheres in metric measure spaces. We first introduce a general setting adapted to the study of asymptotic isoperimetry in a general class of metric measure spaces. We then…
We introduce a concept of tree-graded metric space and we use it to show quasi-isometry invariance of certain classes of relatively hyperbolic groups, to obtain a characterization of relatively hyperbolic groups in terms of their asymptotic…
In this paper, we prove a total curvature estimate of closed hypersurfaces in simply-connected non-positively curved symmetric spaces, and as a corollary, we obtain an isoperimetric inequality for such manifolds.
We initiate a systematic study of quantum properties of finite graphs, namely, quantum asymmetry, quantum symmetry, and quantum isomorphism. We define the Schmidt alternative for a class of graphs, which reveals to be a useful tool for…
Using isoperimetry we obtain new symmetrization inequalities that allow us to provide a unified framework to study Sobolev inequalities in metric spaces. The applications include concentration inequalities, as well as metric versions of the…
We characterize the equivalence and the weak equivalence of Cayley graphs for a finite group $\C{A}$. Using these characterizations, we find enumeration formulae of the equivalence classes and weak equivalence classes of Cayley graphs. As…
We analytically study proximity and distance properties of various kernels and similarity measures on graphs. This helps to understand the mathematical nature of such measures and can potentially be useful for recommending the adoption of…
This article examines the inverse problem for a lossy quantum graph that is internally excited and sensed. In particular, we supply an algorithmic methodology for deducing the topology and geometric structure of the underlying metric graph.…
Recently normalized Laplacian matrices of graphs are studied as density matrices in quantum mechanics. Separability and entanglement of density matrices are important properties as they determine the nonclassical behavior in quantum…
We study free scalar field theory on a graph, which gives rise to a modified version of discrete Green's function on a graph studied in \cite{CY}. We show that this gives rise to a graph invariant, which is closely related to the 2-dim…
We formulate a notion of the quantum automorphism group of a $2$-graph. After some preliminary computations, we define quantum isomorphism between a pair of $2$-graphs. We produce a `non-trivial' example of a pair of $2$-graphs that are not…