Related papers: The spectrum of a random geometric graph is concen…
A powerful framework for studying graphs is to consider them as geometric graphs: nodes are randomly sampled from an underlying metric space, and any pair of nodes is connected if their distance is less than a specified neighborhood radius.…
We investigate spectral properties of quantum graphs in the form of a periodic chain of rings with a connecting link between each adjacent pair, assuming that wave functions at the vertices are matched through conditions manifestly…
In a random key graph (RKG) of $n$ nodes each node is randomly assigned a key ring of $K_n$ cryptographic keys from a pool of $P_n$ keys. Two nodes can communicate directly if they have at least one common key in their key rings. We assume…
We show that the diameter D(G_n) of a random labelled connected planar graph with n vertices is equal to n^{1/4+o(1)}, in probability. More precisely there exists a constant c>0 such that the probability that D(G_n) lies in the interval…
Random graph models are used to describe the complex structure of real-world networks in diverse fields of knowledge. Studying their behavior and fitting properties are still critical challenges, that in general, require model specific…
Given an integer $n$, let $G(n)$ be the number of integer sequences $n-1\ge d_1\ge d_2\ge\dotsb\ge d_n\ge 0$ that are the degree sequence of some graph. We show that $G(n)=(c+o(1))4^n/n^{3/4}$ for some constant $c>0$, improving both the…
We define a correlated random walk (CRW) induced from the time evolution matrix (the Grover matrix) of the Grover walk on a graph $G$, and present a formula for the characteristic polynomial of the transition probability matrix of this CRW…
Given a regular (connected) graph $\Gamma=(X,E)$ with adjacency matrix $A$, $d+1$ distinct eigenvalues, and diameter $D$, we give a characterization of when its distance matrix $A_D$ is a polynomial in $A$, in terms of the adjacency…
We investigate some topological properties of random geometric complexes and random geometric graphs on Riemannian manifolds in the thermodynamic limit. In particular, for random geometric complexes we prove that the normalized counting…
The problem of continuum percolation in dispersions of rods is reformulated in terms of weighted random geometric graphs. Nodes (or sites or vertices) in the graph represent spatial locations occupied by the centers of the rods. The…
Degree distribution, or equivalently called degree sequence, has been commonly used to be one of most significant measures for studying a large number of complex networks with which some well-known results have been obtained. By contrast,…
A spidernet is a graph obtained by adding large cycles to an almost regular tree and considered as an example having intermediate properties of lattices and trees in the study of discrete-time quantum walks on graphs. We introduce the…
A random intersection graph is constructed by assigning independently to each vertex a subset of a given set and drawing an edge between two vertices if and only if their respective subsets intersect. In this paper a model is developed in…
This paper establishes the consistency of spectral approaches to data clustering. We consider clustering of point clouds obtained as samples of a ground-truth measure. A graph representing the point cloud is obtained by assigning weights to…
We study quantum walks on general graphs from the point of view of scattering theory. For a general finite graph we choose two vertices and attach one half line to each. We are interested in walks that proceed from one half line, through…
We establish conditions on sequences of graphs which ensure that the mixing times of the random walks on the graphs in the sequence converge. The main assumption is that the graphs, associated measures and heat kernels converge in a…
Given a dense countable set in a metric space, the infinite random geometric graph is the random graph with the given vertex set and where any two points at distance less than 1 are connected, independently, with some fixed probability. It…
This paper studies how close random graphs are typically to their expectations. We interpret this question through the concentration of the adjacency and Laplacian matrices in the spectral norm. We study inhomogeneous Erd\"os-R\'enyi random…
In this paper, a new measurement to compare two large-scale graphs based on the theory of quantum probability is proposed. An explicit form for the spectral distribution of the corresponding adjacency matrix of a graph is established. Our…
Aldous and Fill conjectured that the maximum relaxation time for the random walk on a connected regular graph with $n$ vertices is $(1+o(1)) \frac{3n^2}{2\pi^2}$. This conjecture can be rephrased in terms of the spectral gap as follows: the…