Spectral lower bounds for the quantum chromatic number of a graph

Elphick, Clive; Wocjan, Paweł

doi:10.1016/j.jcta.2019.06.008

Cited by 9 publications

(24 citation statements)

References 14 publications

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“…which was proved in [4] using different techniques. The proof of the following bound generalises a proof due to Haemers ([7], [8]) from the classical to the quantum chromatic number.…”

Section: Definition 1 (Quantum Coloring) a Quantum C-coloring Of Thementioning

confidence: 97%

“…We use the following alternative characterization of the quantum chromatic number due to [4]. Before stating this characterization, we briefly review the definition of pinching.…”

Section: Definition 1 (Quantum Coloring) a Quantum C-coloring Of Thementioning

confidence: 99%

“…The following theorem is proved as Theorem 1 in [4]. For the sake of completeness, we include a condensed proof below.…”

Section: Definition 1 (Quantum Coloring) a Quantum C-coloring Of Thementioning

confidence: 99%

“…Elphick and Wocjan [4] proved that many spectral lower bounds for χ(G) are also lower bounds for χ q (G), using linear algebra techniques called pinching and twirling. In this paper, we prove that stronger lower bounds on the chromatic number are also lower bound on the quantum chromatic number.…”

Section: Introductionmentioning

confidence: 99%

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Spectral Lower Bounds for the Quantum Chromatic Number of a Graph – Part II

Wocjan¹,

Elphick²,

Darbari³

2020

Electron. J. Combin.

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Hoffman proved that a graph $G$ with eigenvalues $\mu_1 \geqslant \cdots \geqslant \mu_n$ and chromatic number $\chi(G)$ satisfies: \[\chi \geqslant 1 + \kappa\] where $\kappa$ is the smallest integer such that \[\mu_1 + \sum_{i=1}^{\kappa} \mu_{n+1-i} \leqslant 0.\] We strengthen this well known result by proving that $\chi(G)$ can be replaced by the quantum chromatic number, $\chi_q(G)$, where for all graphs $\chi_q(G) \leqslant \chi(G)$ and for some graphs $\chi_q(G)$ is significantly smaller than $\chi(G)$. We also prove a similar result, and investigate implications of these inequalities for the quantum chromatic number of various classes of graphs, which improves many known results. For example, we demonstrate that the Kneser graph $KG_{p,2}$ has $\chi_q = \chi = p - 2$.

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“…which was proved in [4] using different techniques. The proof of the following bound generalises a proof due to Haemers ([7], [8]) from the classical to the quantum chromatic number.…”

Section: Definition 1 (Quantum Coloring) a Quantum C-coloring Of Thementioning

confidence: 97%

“…We use the following alternative characterization of the quantum chromatic number due to [4]. Before stating this characterization, we briefly review the definition of pinching.…”

Section: Definition 1 (Quantum Coloring) a Quantum C-coloring Of Thementioning

confidence: 99%

“…The following theorem is proved as Theorem 1 in [4]. For the sake of completeness, we include a condensed proof below.…”

Section: Definition 1 (Quantum Coloring) a Quantum C-coloring Of Thementioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Spectral Lower Bounds for the Quantum Chromatic Number of a Graph – Part II

Wocjan¹,

Elphick²,

Darbari³

2020

Electron. J. Combin.

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“…Wocjan and Elphick [5] strengthened (3) by proving that the Ando and Lin bound is a lower bound for the quantum chromatic number, χ q (G), with arbitrary Hermitian weight matrices. Wocjan and Elphick [19] further strengthened (3) by proving that the Kolotilina and Lima et al bounds are lower bounds for the vectorial chromatic number, χ vect (G) = ϑ + (G) , again with arbitrary Hermitian weight matrices.…”

Section: Spectral Lower Bounds For Chromatic Numbersmentioning

confidence: 99%

More tales of Hoffman: bounds for the vector chromatic number of a graph

Wocjan

Elphick²,

Anekstein³

2020

Discuss. Math. Graph Theory

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Let χ(G) denote the chromatic number of a graph and χ v (G) denote the vector chromatic number. For all graphs χ v (G) ≤ χ(G) and for some graphs χ v (G) χ(G). Galtman proved that Hoffman's well-known lower bound for χ(G) is in fact a lower bound for χ v (G). We prove that two more spectral lower bounds for χ(G) are also lower bounds for χ v (G). We then use one of these bounds to derive a new characterization of χ v (G).

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New eigenvalue bound for the fractional chromatic number

Guo,

Spiro

2023

Journal of Graph Theory

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Given a graph , we let denote the sum of the squares of the positive eigenvalues of the adjacency matrix of , and we similarly define . We prove that and thus strengthen a result of Ando and Lin, who showed the same lower bound for the chromatic number . We in fact show a stronger result wherein we give a bound using the eigenvalues of and whenever has a homomorphism to an edge‐transitive graph . Our proof utilizes ideas motivated by association schemes.

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Spectral lower bounds for the quantum chromatic number of a graph

Cited by 9 publications

References 14 publications

Spectral Lower Bounds for the Quantum Chromatic Number of a Graph – Part II

Spectral Lower Bounds for the Quantum Chromatic Number of a Graph – Part II

More tales of Hoffman: bounds for the vector chromatic number of a graph

New eigenvalue bound for the fractional chromatic number

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