Related papers: Quantum graphs which optimize the spectral gap

In 2019, Aterias et al. constructed pairs of quantum isomorphic, non-isomorphic graphs from linear constraint systems. This article deals with quantum automorphisms and quantum isomorphisms of colored versions of those graphs. We show that…

We study topological Poincar\'e type inequalities on general graphs. We characterize graphs satisfying such inequalities and then turn to the best constants in these inequalities. Invoking suitable metrics we can interpret these constants…

We investigate graphs that can be disconnected into small components by removing a vanishingly small fraction of their vertices. We show that when a quantum network is described by such a graph, the network is efficiently controllable, in…

Not necessarily self-adjoint quantum graphs -- differential operators on metric graphs -- are considered. Assume in addition that the underlying metric graph possesses an automorphism (symmetry) $ \mathcal P $. If the differential operator…

Many computational problems are unchanged under some symmetry operation. In classical machine learning, this can be reflected with the layer structure of the neural network. In quantum machine learning, the ansatz can be tuned to correspond…

In this paper, we characterize all graphs with eigenvectors of the signless Laplacian and adjacency matrices with components equal to $\{- 1, 0, 1\}.$ We extend the graph parameter max $k$-cut to square matrices and prove a general sharp…

We address the "inverse problem" for discrete geometry, which consists in determining whether, given a discrete structure of a type that does not in general imply geometrical information or even a topology, one can associate with it a…

In this paper, we initiate the study of quantum algorithms in the Graph Drawing research area. We focus on two foundational drawing standards: 2-level drawings and book layouts. Concerning $2$-level drawings, we consider the problems of…

For a given graph $G$, we aim to determine the possible realizable spectra for a generalized (or sometimes referred to as a weighted) Laplacian matrix associated with $G$. This new specialized inverse eigenvalue problem is considered for…

We determine the quantum automorphism groups of finite graphs. These are quantum subgroups of the quantum permutation groups defined by Wang. The quantum automorphism group is a stronger invariant for finite graphs than the usual one. We…

Quantum computers hold great promise, but it remains a challenge to find efficient quantum circuits that solve interesting computational problems. We show that finding optimal quantum circuits is essentially equivalent to finding the…

We study two different types of gluing for graphs: interface (obtained by choosing a common subgraph as the gluing component) and bridge gluing (obtained by adding a set of edges to the given subgraphs). We introduce formulae for computing…

In this brief paper we present some results on creating and manipulating spectral gaps for a (regular) quantum graph by inserting appropriate internal structures into its vertices. Complete proofs and extensions of the results are planned…

Trace formulas play a central role in the study of spectral geometry and in particular of quantum graphs. The basis of our work is the result by Kurasov which links the Euler characteristic $\chi$ of metric graphs to the spectrum of their…

There is a deep and interesting connection between the topological properties of a graph and the behaviour of the dynamical system defined on it. We analyse various kind of graphs, with different contrasting connectivity or degree…

We present an efficient quantum algorithm for some independent set problems in graph theory, based on non-abelian adiabatic mixing. We illustrate the performance of our algorithm with analysis and numerical calculations for two different…

Quantum geometry on a discrete set means a directed graph with a weight associated to each arrow defining the quantum metric. However, these `lattice spacing' weights do not have to be independent of the direction of the arrow. We use this…

We consider a non-compact Riemannian periodic manifold such that the corresponding Laplacian has a spectral gap. By continuously perturbing the periodic metric locally we can prove the existence of eigenvalues in a gap. A lower bound on the…

We propose a novel variational method for solving the sub-graph isomorphism problem on a gate-based quantum computer. The method relies (1) on a new representation of the adjacency matrices of the underlying graphs, which requires a number…

We generalize the concept of token graphs to obtain supertoken graphs. In the latter case, there can be more than one token in a vertex. We formally define supertoken graphs and establish their basic properties. Moreover, we provide some…