Related papers: On the quantum chromatic number of Hamming and gen…

The study of quantum chromatic numbers of graphs is a hot research topic in recent years. However, the infinite family of graphs with known quantum chromatic numbers are rare, as far as we know, the only known such graphs (except for…

Quantum graphs are an operator space generalization of classical graphs that have emerged in different branches of mathematics including operator theory, non-commutative topology and quantum information theory. In this paper, we obtain…

The determination of the quantum chromatic number of graphs has attracted considerable attention recently. However, there are few families of graphs whose quantum chromatic numbers are determined. A notable exception is the family of…

The smallest eigenvalues of (distance-j) Hamming graphs with distance parameter j at least half the length were completely determined by Brouwer et al. (2018). In the present work, we address the complementary regime, namely distances j…

We investigate the notion of quantum chromatic number of a graph, which is the minimal number of colours necessary in a protocol in which two separated provers can convince an interrogator with certainty that they have a colouring of the…

We compute the exact value of the quantum chromatic numbers of Hadamard graphs of order $n=2^N$ for $N$ a multiple of $4$ using the upper bound derived by Avis, Hasegawa, Kikuchi, and Sasaki, as well as an application of the Hoffman-like…

The quantum chromatic number, a generalization of the chromatic number, was first defined in relation to the non-local quantum coloring game. We generalize the former by defining the quantum $k$-distance chromatic number $\chi_{kq}(G)$ of a…

We study quantum analogs of graph colorings and chromatic number. Initially defined via an interactive protocol, quantum colorings can also be viewed as a natural operator relaxation of graph coloring. Since there is no known algorithm for…

The largest known gap between quantum and classical chromatic number of graphs, obtained via quantum protocols for colouring Hadamard graphs based on the Deutsch--Jozsa algorithm and the quantum Fourier transform, is exponential. We put…

We develop further the new versions of quantum chromatic numbers of graphs introduced by the first and fourth authors. We prove that the problem of computation of the commuting quantum chromatic number of a graph is solvable by an SDP…

First, I introduce quantum graph theory. I also discuss a known lower bound on the independence numbers and derive from it an upper bound on the chromatic numbers of quantum graphs. Then, I construct a family of quantum graphs that can be…

A lower bound on the chromatic number of a graph is derived by majorization of spectra of weighted adjacency matrices. These matrices are given by Hadamard products of the adjacency matrix and arbitrary Hermitian matrices.

This paper deals with graph colouring games, an example of pseudo-telepathy, in which two provers can convince a verifier that a graph $G$ is $c$-colourable where $c$ is less than the chromatic number of the graph. They win the game if they…

The quantum chromatic number, $\chi_q(G)$, of a graph $G$ was originally defined as the minimal number of colors necessary in a quantum protocol in which two provers that cannot communicate with each other but share an entangled state can…

Motivated by non-local games and quantum coloring problems, we introduce a graph homomorphism game between quantum graphs and classical graphs. This game is naturally cast as a "quantum-classical game"--that is, a non-local game of two…

Hoffman proved that a graph $G$ with adjacency eigenvalues $\lambda_1\geq \cdots \geq \lambda_n$ and chromatic number $\chi(G)$ satisfies $\chi(G)\geq 1+\kappa,$ where $\kappa$ is the smallest integer such that…

We show that quantum graph parameters for finite, simple, undirected graphs encode winning strategies for all possible synchronous non-local games. Given a synchronous game $\mathcal{G}=(I,O,\lambda)$ with $|I|=n$ and $|O|=k$, we…

Hoffman proved that a graph $G$ with eigenvalues $\mu_1 \ge \ldots \ge \mu_n$ and chromatic number $\chi(G)$ satisfies: \[ \chi \ge 1 + \kappa \] where $\kappa$ is the smallest integer such that \[ \mu_1 + \sum_{i=1}^{\kappa} \mu_{n+1-i}…

A homomorphism from a graph $X$ to a graph $Y$ is an adjacency preserving mapping $f:V(X) \rightarrow V(Y)$. We consider a nonlocal game in which Alice and Bob are trying to convince a verifier with certainty that a graph $X$ admits a…

A lambda colouring (or $L(2,1)-$colouring) of a graph is an assignment of non-negative integers (with minimum assignment $0$) to its vertices such that the adjacent vertices must receive integers at least two apart and vertices at distance…