Related papers: Sharp diameter bound on the spectral gap for quant…
The existence of a strong spectral gap for quotients $\Gamma\bs G$ of noncompact connected semisimple Lie groups is crucial in many applications. For congruence lattices there are uniform and very good bounds for the spectral gap coming…
Motivated by a recent application of quantum graphs to model the anomalous Hall effect we discuss quantum graphs the vertices of which exhibit a preferred orientation. We describe an example of such a vertex coupling and analyze the…
We study the transmission of a quantum particle along a straight input--output line to which a graph $\Gamma$ is attached at a point. In the point of contact we impose a singularity represented by a certain properly chosen scale-invariant…
We give a proof for sharp estimate for the number of spanning trees using linear algebra and generalize this bound to multigraphs. In addition, we show that this bound is tight for complete graphs. In addition, we give estimates for number…
The degree-diameter problem seeks to find the maximum possible order of a graph with a given (maximum) degree and diameter. It is known that graphs attaining the maximum possible value (the Moore bound) are extremely rare, but much activity…
The symmetries of a finite graph are described by its automorphism group; in the setting of Woronowicz's quantum groups, a notion of a quantum automorphism group has been defined by Banica capturing the quantum symmetries of the graph. In…
In this paper we consider the concept of preintersection numbers of a graph. These numbers are determined by the spectrum of the adjacency matrix of the graph, and generalize the intersection numbers of a distance-regular graph. By using…
The quantum black hole model with a self-gravitating spherically symmetric thin dust shell as a source is considered. The shell Hamiltonian constraint is written and the corresponding Schroedinger equation is obtained. This equation…
The inertia bound gives an upper bound on the independence number of a graph by considering the inertia of matrices corresponding to the graph. The bound is known to be tight for graphs on 10 or fewer vertices as well as for all perfect…
We study a family of closed quantum graphs described by one singular vertex of order n=4. By suitable choice of the parameters specifying the singular vertex, we can construct a closed sequence of paths in the parameter space that…
We construct a family of finite special 2-groups which have commuting graph of increasing diameter
For any given natural number $k$, this paper gives upper bounds on the radius of a packing of a complete hyperbolic surface of finite area by $k$ equal-radius disks in terms of the surface's topology. We show that the bounds given here are…
We analytically calculate the energy spectrum of a circular graphene quantum dot with radius R subjected to a perpendicular magnetic field B by applying the infinite-mass boundary condition. We can retrieve well-known limits for the cases…
We show there is an upper bound on the diameter of a closed, hyperbolic 3-manifold in terms of the length of any presentation of its fundamental group.
In this paper we analyze the spectral gap of a weighted graph which is the difference between the smallest positive and largest negative eigenvalue of its adjacency matrix. Such a graph can represent e.g. a chemical organic molecule. Our…
We show that a graph with $n$ vertices and vertex cover of size $k$ has at most $4^k + n$ potential maximal cliques. We also show that for each positive integer $k$, there exists a graph with vertex cover of size $k$, $O(k^2)$ vertices, and…
Quantum graphs have recently emerged as models of nonlinear optical, quantum confined systems with exquisite topological sensitivity and the potential for predicting structures with an intrinsic, off-resonance response approaching the…
A graph is minimally $k$-connected ($k$-edge-connected) if it is $k$-connected ($k$-edge-connected) and deleting arbitrary chosen edge always leaves a graph which is not $k$-connected ($k$-edge-connected). A classic result of minimally…
Random quantum circuits are a central concept in quantum information theory with applications ranging from demonstrations of quantum computational advantage to descriptions of scrambling in strongly-interacting systems and black holes. The…
Recently, the work on quantum automorphism groups of graphs has seen renewed progress, which we expand in this paper. Quantum symmetry is a richer notion of symmetry than the classical symmetries of a graph. In general, it is non-trivial to…