Chromatic numbers, Sabidussi's Theorem and Hedetniemi's conjecture for non-commutative graphs

Kim, Se-Jin; Mehta, Arthur

doi:10.1016/j.laa.2019.08.002

Cited by 6 publications

(2 citation statements)

References 8 publications

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“…Let S = span {I, E ij : i = j} ⊆ M n , which is a quantum graph on M n . It is known [27] that χ((S, M n , M n )) = n. Here, we will show that χ qc ((S, M n )) = n as well, which shows that χ t ((S, M n )) = n for any t ∈ {loc, q, qa, qc}.…”

Section: Coloring Quantum Graphsmentioning

confidence: 53%

The quantum-to-classical graph homomorphism game

Brannan¹,

Ganesan²,

Harris³

2020

Preprint

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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 "quantumclassical game"-that is, a non-local game of two players involving quantum questions and classical answers. This game generalizes the graph homomorphism game between classical graphs. We show that winning strategies in the various quantum models for the game directly generalize the notion of non-commutative graph homomorphisms due to D. Stahlke [44]. Moreover, we present a game algebra in this context that generalizes the game algebra for graph homomorphisms given by J.W. Helton, K. Meyer, V.I. Paulsen and M. Satriano [22]. We also demonstrate explicit quantum colorings of all quantum complete graphs, yielding the surprising fact that the algebra of the 4-coloring game for a quantum graph is always non-trivial, extending a result of [22].

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Section: Coloring Quantum Graphsmentioning

confidence: 53%

The quantum-to-classical graph homomorphism game

Brannan¹,

Ganesan²,

Harris³

2020

Preprint

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“…These so called non-commutative graphs have since drawn interest in connection with quantum channels but also independently of any application. Several classical graph definitions and results carry over to non-commutative graphs, including homomorphisms [5,25,27,30], chromatic numbers [17,19,27], Ramsey and Turán theorems [31,32], asymptotic spectrum [20], a Haemers bound [15], and connectivity [6]. It can happen that there are multiple ways to generalize a particular concept: [3] presents two generalizations of ϑ distinct from the one in [10] (though possibly the same as each other).The present work investigates a weighted version of the θ of [10], generalizing most of the results from [16].…”

mentioning

confidence: 99%

Weighted theta functions for non-commutative graphs

Stahlke¹

2021

Preprint

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Grötschel, Lovász, and Schrijver generalized the Lovász ϑ function by allowing a weight for each vertex. We provide a similar generalization of Duan, Severini, and Winter's θ on non-commutative graphs. While the classical theory involves a weight vector assigning a non-negative weight to each vertex, the non-commutative theory uses a positive semidefinite weight matrix. The classical theory is recovered in the case of diagonal weight matrices.Most of Grötschel, Lovász, and Schrijver's results generalize to non-commutative graphs. In particular, we generalize the inequality ϑ(G, w)ϑ(G, x) ≥ w, x with some modification needed due to non-commutative graphs having a richer notion of complementation. Similar to the classical case, facets of the theta body correspond to cliques and if the theta body anti-blocker is finitely generated then it is equal to the non-commutative generalization of the clique polytope.We propose two definitions for non-commutative perfect graphs, equivalent for classical graphs but inequivalent for non-commutative graphs. [22] introduced the ϑ function of a graph as an upper bound on the Shannon capacity -the independence number regularized under the strong graph product. The ϑ quantity is an upper bound on independence number, a lower bound on fractional chromatic number, and is multiplicative under the strong and the disjunctive graph products. It is a semidefinite program, hence efficiently computable both in theory and in practice. It is monotone under graph homomorphisms [8]; in fact its bound on independence and chromatic number follow from this. I. INTRODUCTION LovászFurther insight into ϑ is gained by allowing vertices to be weighted [16,18]. Weights are basically equivalent to duplicating vertices [18] except that weights don't have to be whole numbers. Aside from only being defined for non-negative weights, the weighted ϑ of a graph resembles a norm on the weight vector: it scales linearly and is convex. In that language, ϑ of the complement graph is the dual norm. The set of weights w for which ϑ(G, w) ≤ 1 is investigated in [16], where facets of this convex body are shown to correspond to clique constraints. This set is polyhedral if and only if the graph is perfect.Lovász's bound can be adapted to quantum channels via a suitable generalization of graphs where an operator subspace takes the place of the adjacency matrix [10]. These so called non-commutative graphs have since drawn interest in connection with quantum channels but also independently of any application. Several classical graph definitions and results carry over to non-commutative graphs, including homomorphisms [5,25,27,30], chromatic numbers [17,19,27], Ramsey and Turán theorems [31,32], asymptotic spectrum [20], a Haemers bound [15], and connectivity [6]. It can happen that there are multiple ways to generalize a particular concept: [3] presents two generalizations of ϑ distinct from the one in [10] (though possibly the same as each other).The present work investigates a weighted version of the θ of [10], generalizi...

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Connectivity for quantum graphs

Chávez-Domínguez

Swift

2021

Linear Algebra and its Applications

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Chromatic numbers, Sabidussi's Theorem and Hedetniemi's conjecture for non-commutative graphs

Cited by 6 publications

References 8 publications

The quantum-to-classical graph homomorphism game

The quantum-to-classical graph homomorphism game

Weighted theta functions for non-commutative graphs

Connectivity for quantum graphs

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