Related papers: Vertices cannot be hidden from quantum spatial sea…
The spectrum of a network or graph $G=(V,E)$ with adjacency matrix $A$, consists of the eigenvalues of the normalized Laplacian $L= I - D^{-1/2} A D^{-1/2}$. This set of eigenvalues encapsulates many aspects of the structure of the graph,…
We study numerically and analytically the spectrum of incidence matrices of random labeled graphs on N vertices : any pair of vertices is connected by an edge with probability p. We give two algorithms to compute the moments of the…
A graph is said to be $\mathcal{H}(n, \Delta)$-universal if it contains every graph on $n$ vertices with maximum degree at most $\Delta$. Using a `matching-based' embedding technique introduced by Alon and F\"uredi, Dellamonica, Kohayakawa,…
A matching M is a dominating induced matching of a graph, if every edge of the graph is either in $M$ or has a common end-vertex with exactly one edge in $M$. The concept of complete dominating induced matching is introduced as graphs where…
We construct a new quantum algorithm for the graph collision problem; that is, the problem of deciding whether the set of marked vertices contains a pair of adjacent vertices in a known graph G. The query complexity of our algorithm is…
We consider three different models of sparse random graphs:~undirected and directed Erd\H{o}s-R\'{e}nyi graphs, and random bipartite graph with an equal number of left and right vertices. For such graphs we show that if the edge…
This paper provides a construction method of the nearest graph Laplacian to a matrix identified from measurement data of graph Laplacian dynamics that include biochemical systems, synchronization systems, and multi-agent systems. We…
We prove that for $k+1\geq 3$ and $c>(k+1)/2$ w.h.p. the random graph on $n$ vertices, $cn$ edges and minimum degree $k+1$ contains a (near) perfect $k$-matching. As an immediate consequence we get that w.h.p. the $(k+1)$-core of $G_{n,p}$,…
In graph analysis, a classic task consists in computing similarity measures between (groups of) nodes. In latent space random graphs, nodes are associated to unknown latent variables. One may then seek to compute distances directly in the…
Proximity graph-based methods have emerged as a leading paradigm for approximate nearest neighbor (ANN) search in the system community. This paper presents fresh insights into the theoretical foundation of these methods. We describe an…
In network analysis and graph mining, closeness centrality is a popular measure to infer the importance of a vertex. Computing closeness efficiently for individual vertices received considerable attention. The NP-hard problem of group…
Although strongly regular graphs and the hypercube are not complete, they are "sufficiently complete" such that a randomly walking quantum particle asymptotically searches on them in the same $\Theta(\sqrt{N})$ time as on the complete…
In the paper, we investigated the influence of local and global interaction on the efficiency of continuous-time quantum spatial search. To do so, we analyzed numerically chimera graph, which is defined as 2D grid with each node replaced by…
We study quantum algorithms for spatial search on finite dimensional grids. Patel et al. and Falk have proposed algorithms based on a quantum walk without a coin, with different operators applied at even and odd steps. Until now, such…
We study the average probability that a discrete-time quantum walk finds a marked vertex on a graph. We first show that, for a regular graph, the spectrum of the transition matrix is determined by the weighted adjacency matrix of an…
Very sparse random graphs are known to typically be singular (i.e., have singular adjacency matrix), due to the presence of "low-degree dependencies'' such as isolated vertices and pairs of degree-1 vertices with the same neighbourhood. We…
In this thesis, we study Laplacian eigenfunctions on metric graphs, also known as quantum graphs. We restrict the discussion to standard quantum graphs. These are finite connected metric graphs with functions that satisfy Neumann vertex…
We consider the problem of optimal recovery of true ranking of $n$ items from a randomly chosen subset of their pairwise preferences. It is well known that without any further assumption, one requires a sample size of $\Omega(n^2)$ for the…
Our goal is to efficiently compute low-dimensional latent coordinates for nodes in an input graph -- known as graph embedding -- for subsequent data processing such as clustering. Focusing on finite graphs that are interpreted as uniform…
We consider an inverse problem for a finite graph $(X,E)$ where we are given a subset of vertices $B\subset X$ and the distances $d_{(X,E)}(b_1,b_2)$ of all vertices $b_1,b_2\in B$. The distance of points $x_1,x_2\in X$ is defined as the…