Related papers: Generating isospectral but not isomorphic quantum …
In this monography, it is proposed to consider the concepts of spectra of edge cuts and edge cycles of a graph as a basic mathematical structure for solving the problem of graph isomorphism. An edge cut is defined by an edge and the…
The paper deals with some spectral properties of (mostly infinite) quantum and combinatorial graphs. Quantum graphs have been intensively studied lately due to their numerous applications to mesoscopic physics, nanotechnology, optics, and…
We study a set of scattering matrices of quantum graphs containing minimal number of passbands, i.e., maximal number of zero elements. The cases of even and odd vertex degree are considered. Using a solution of inverse scattering problem,…
Construction of graphs with equal eigenvalues (co-spectral graphs) is an interesting problem in spectral graph theory. Seidel switching is a well-known method for generating co-spectral graphs. From a matrix theoretic point of view, Seidel…
Determining whether two graphs are structurally identical is a fundamental problem with applications spanning mathematics, computer science, chemistry, and network science. Despite decades of study, graph isomorphism remains a challenging…
We construct models of exactly solvable two-particle quantum graphs with certain non-local two-particle interactions, establishing appropriate boundary conditions via suitable self-adjoint realisations of the two-particle Laplacian. Showing…
We prove that for every pair of quantum isomorphic graphs, their block trees and their block graphs are isomorphic, and that such an isomorphism can be chosen so that the corresponding blocks are quantum isomorphic -- in particular,…
We show that any two Hadamard graphs on the same number of vertices are quantum isomorphic. This follows from a more general recipe for showing quantum isomorphism of graphs arising from certain association schemes. The main result is built…
The paper is concerned with the number of open gaps in spectra of periodic quantum graphs. The well-known conjecture by Bethe and Sommerfeld (1933) says that the number of open spectral gaps for a system periodic in more than one direction…
We summarize different approaches to the theory of quantum graphs and provide several ways to construct concrete examples. First, we classify all undirected quantum graphs on the quantum space $M_2$. Secondly, we apply the theory of…
Two graphs having the same spectrum are said to be cospectral. Two graphs such that the absolute values of their nonzero eigenvalues coincide are singularly cospectral graphs. Cospectrality implies singular cospectrality, but the converse…
We present a strong connection between quantum information and quantum permutation groups. Specifically, we define a notion of quantum isomorphisms of graphs based on quantum automorphisms from the theory of quantum groups, and then show…
According to a recent conjecture, isospectral objects have different nodal count sequences. We study generalized Laplacians on discrete graphs, and use them to construct the first non-trivial counter-examples to this conjecture. In…
Three new graph invariants are introduced which may be measured from a quantum graph state and form examples of a framework under which other graph invariants can be constructed. Each invariant is based on distinguishing a different number…
We undertake a study of the notion of a quantum graph over arbitrary finite-dimensional $C^*$-algebras $B$ equipped with arbitrary faithful states. Quantum graphs are realised principally as either certain operators on $L^2(B)$, the quantum…
We quantize the regularity properties of classical graphs that determine spin models for singly-generated Yang-Baxter planar algebras, including the Kauffman polynomial, and construct explicit examples. A source of examples comes from…
Spectral problems are considered generated by the Sturm-Liouville equation on connected simple equilateral graphs with the Neumann and Dirichlet boundary conditions at the pendant vertices and continuity and Kirchhoff's conditions at the…
The state-of-the-art solvers for the graph isomorphism problem can readily solve generic instances with tens of thousands of vertices. Indeed, experiments show that on inputs without particular combinatorial structure the algorithms scale…
We introduce a new model for investigating spectral properties of quantum graphs, a quantum circulant graph. Circulant graphs are the Cayley graphs of cyclic groups. Quantum circulant graphs with standard vertex conditions maintain…
The graph isomorphism problem asks whether two graphs are identical up to vertex relabeling. While the exact problem admits quasi-polynomial-time classical algorithms, many applications in molecular comparison, noisy network analysis, and…