相关论文: A Quantum Observable for the Graph Isomorphism Pro…

Let $G$ be a group. The BCI problem asks whether two Haar graphs of $G$ are isomorphic if and only if they are isomorphic by an element of an explicit list of isomorphisms. We first generalize this problem in a natural way and give a…

We consider a method popular in the literature of associating a two-step nilpotent Lie algebra with a finite simple graph. We prove that the two-step nilpotent Lie algebras associated with two graphs are Lie isomorphic if and only if the…

Graph partitioning, or the dividing of a graph into two or more parts based on certain conditions, arises naturally throughout discrete mathematics, and problems of this kind have been studied extensively. In the 1990s, Ando conjectured…

We consider a connected graph $\Gamma$ as a coarse space and prove that $\Gamma$ admits a 2-selector if and only if $\Gamma$ is either bounded or coarsely equivalent to $\mathbb{N}$ or $\mathbb{Z}$. We apply this result to geodesic metric…

A graph of order $n>3$ is called {switching separable} if its modulo-2 sum with some complete bipartite graph on the same set of vertices is divided into two mutually independent subgraphs, each having at least two vertices. We prove the…

A connected 3-valent plane graph, whose faces are $q$- or 6-gons only, is called a {\em graph $q_n$}. We classify all graphs $4_n$, which are isometric subgraphs of a $m$-hypercube $H_m$.

Quantum observables can be identified with vector fields on the sphere of normalized states. Consequently, the uncertainty relations for quantum observables become geometric statements. In the Letter the familiar uncertainty relation…

Normalized Laplacian matrices of graphs have recently been studied in the context of quantum mechanics as density matrices of quantum systems. Of particular interest is the relationship between quantum physical properties of the density…

We investigate the graph isomorphism (GI) in some cospectral networks. Two graph are isomorphic when they are related to each other by a relabeling of the graph vertices. We want to investigate the GI in two scalable (n + 2)-regular graphs…

The Subgraph Isomorphism problem asks, given a host graph G on n vertices and a pattern graph P on k vertices, whether G contains a subgraph isomorphic to P. The restriction of this problem to planar graphs has often been considered. After…

We investigate classical and quantum physics-based algorithms for solving the graph isomorphism problem. Our work integrates and extends previous work by Gudkov et al. (cond-mat/0209112) and by Rudolph (quant-ph/0206068). Gudkov et al.…

We investigate the bottom of the spectra of infinite quantum graphs, i.e., Laplace operators on metric graphs having infinitely many edges and vertices. We introduce a new definition of the isoperimetric constant for quantum graphs and then…

An edge-colored directed graph is \emph{observable} if an agent that moves along its edges is able to determine his position in the graph after a sufficiently long observation of the edge colors. When the agent is able to determine his…

Graph states are well-entangled quantum states that are defined based on a graph. Of course, if two graphs are isomorphic their associated states are the same. Also, we know local operations do not change the entanglement of quantum states.…

In 2003, van Dam and Haemers posed a fundamental question in spectral graph theory: does there exist a ``sensible'' matrix whose spectrum determines a random graph up to isomorphism? This paper introduces the class of {\em natural graph…

The quantum lens spaces form a natural and well-studied class of noncommutative spaces which can be subjected to classification using algebraic invariants by drawing on the fully developed classification theory of unital graph…

The complexity of the graph isomorphism problem for trapezoid graphs has been open over a decade. This paper shows that the problem is GI-complete. More precisely, we show that the graph isomorphism problem is GI-complete for comparability…

We prove that a graph whose chromatic number is 2 is a homotopy test graph. We also prove that there is a graph $K$ with two involutions $\gamma_1$ and $\gamma_2$ such that $(K,\gamma_1)$ is a Stiefel-Whitney test graph, but $(K,\gamma_2)$…

We study the problem of finding a copy of a specific induced subgraph on inhomogeneous random graphs with infinite variance power-law degrees. We provide a fast algorithm that finds a copy of any connected graph $H$ on a fixed number of $k$…

Let $\A$ and $\B$ be operator algebras with $c_0$-isomorphic diagonals and let $\K$ denote the compact operators. We show that if $\A\otimes\K$ and $\B\otimes\K$ are isometrically isomorphic, then $\A$ and $\B$ are isometrically isomorphic.…