Related papers: Spectral gap for quantum graphs and their connecti…
For a finite graph, we establish natural isomorphisms between eigenspaces of a Laplace operator acting on functions on the edges and eigenspaces of a transfer operator acting on functions on one-sided infinite non-backtracking paths.…
We derive a sharp criterion on the spectra of periodic discrete Schr\"odinger operators acting on connected periodic lattices: the measure of the spectrum goes to zero as the coupling constant goes to infinity if and only if there is no…
Understanding efficient modifications to improve network functionality is a fundamental problem of scientific and industrial interest. We study the response of network dynamics against link modifications on a weakly connected directed graph…
This article deals with the spectra of Laplacians of weighted graphs. In this context, two objects are of fundamental importance for the dynamics of complex networks: the second eigenvalue of such a spectrum (called algebraic connectivity)…
Complex networks or graphs are ubiquitous in sciences and engineering: biological networks, brain networks, transportation networks, social networks, and the World Wide Web, to name a few. Spectral graph theory provides a set of useful…
Quantum network is fragile to disturbances when qubits are transmitted through quantum channel. Reliability is an essential requirement for a quantum network and even the future quantum internet. A metric is needed to describe the…
The adjacency operator of a graph has a spectrum and a class of scalar-valued spectral measures which have been systematically analyzed; it also has a spectral multiplicity function which has been less studied. The first purpose of this…
The distance matrix of a connected graph is the symmetric matrix with columns and rows indexed by the vertices and entries that are the pairwise distances between the corresponding vertices. We give a construction for graphs which differ in…
The famous question of Mark Kac "Can one hear the shape of a drum?" addressing the unique connection between the shape of a planar region and the spectrum of the corresponding Laplace operator can be legitimately extended to scattering…
Large-scale graphs are widely used to represent object relationships in many real world applications. The occurrence of large-scale graphs presents significant computational challenges to process, analyze, and extract information. Graph…
We show that if a graph is k-edge-connected, and we adjoin to it another graph satisfying a "contracted diameter less or equal to 2" condition, with minimal degree greater or equal to k, and some natural hypothesis on the edges connecting…
The two-dimensional Dirac operator describes low-energy excitations in graphene. Different choices for the boundary conditions give rise to qualitative differences in the spectrum of the resulting operator. For a family of boundary…
We study the influence of certain geometric perturbations on the spectra of self-adjoint Schr\"odinger operators on compact metric graphs. Results are obtained for permutation invariant vertex conditions, which, amongst others, include…
We show that graphs, networks and other related discrete model systems carry a natural supersymmetric structure, which, apart from its conceptual importance as to possible physical applications, allows to derive a series of spectral…
In the past decades, graphs that are determined by their spectrum have received more attention, since they have been applied to several fields, such as randomized algorithms, combinatorial optimization problems and machine learning. An…
Inspired by notorious combinatorial optimization problems on graphs, in this paper we consider a series of related problems defined using a metric space and topology determined by a graph. Particularly, we present the Independent Set,…
A continuous-time quantum walk is modelled using a graph. In this short paper, we provide lower bounds on the size of a graph that would allow for some quantum phenomena to occur. Among other things, we show that, in the adjacency matrix…
We prove that the spectral gap of a finite planar graph $X$ is bounded by $\lambda_1(X)\le C(\frac{\log(\diam X)}{\diam X})^2$ where $C$ depends only on the degree of $X$. We then give a sequence of such graphs showing the the above…
In this paper a tight lower bound for algebraic connectivity of graphs (second smallest eigenvalue of the Laplacian matrix of the graph) based on connection-graph-stability method is introduced. The connection-graph-stability score for each…
The Spectral Excess Theorem (SPET) for distance-regular graphs states that a regular (connected) graph is distance-regular if and only if its spectral-excess equals its average excess. Recently, some local or global approaches to the SPET…