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Quantum isomorphism of graphs from association schemes
Combinatorics · Submitted 10 Sep 2022
abstract compiled · 2026-09-23 · no language-model review
The authors' main claim We show that any two Hadamard graphs on the same number of vertices are quantum isomorphic.
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The abstract, sentence by sentence
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 from three tools. A remarkable recent result of Man\v{c}inska and Roberson shows that graphs $G$ and $H$ are quantum isomorphic if and only if, for any planar graph $F$, the number of graph homomorphisms from $F$ to $G$ is equal to the number of graph homomorphisms from $F$ to $H$. A generalization of partition functions called "scaffolds" affords some basic reduction rules such as series-parallel reduction and can be applied to counting homomorphisms. The final tool is the classical theorem of Epifanov showing that any plane graph can be reduced to a single vertex and no edges by extended series-parallel reductions and Delta-Wye transformations. This last sort of transformation is available to us in the case of exactly triply regular association schemes. The paper includes open problems and directions for future research.
background contribution method result interpretation main claim
| # | sentence | role | the authors' own work | states the main conclusion | strength |
|---|---|---|---|---|---|
| s0 | We show that any two Hadamard graphs on the same number of vertices are quantum isomorphic. | result 0.97 | 0.94 | 0.81 | asserted 1.00 |
| s1 | This follows from a more general recipe for showing quantum isomorphism of graphs arising from certain association schemes. | contribution 0.43 | 0.75 | 0.08 | asserted 1.00 |
| s2 | The main result is built from three tools. | method 0.86 | 0.34 | 0.03 | asserted 0.99 |
| s3 | A remarkable recent result of Man\v{c}inska and Roberson shows that graphs $G$ and $H$ are quantum isomorphic if and only if, for any planar graph $F$, the number of graph homomorphisms from $F$ to $G$ is equal to the number of graph homomorphisms from $F$ to $H$. | background 0.92 | 0.02 | 0.04 | asserted 1.00 |
| s4 | A generalization of partition functions called "scaffolds" affords some basic reduction rules such as series-parallel reduction and can be applied to counting homomorphisms. | method 0.87 | 0.21 | 0.02 | asserted 0.85 |
| s5 | The final tool is the classical theorem of Epifanov showing that any plane graph can be reduced to a single vertex and no edges by extended series-parallel reductions and Delta-Wye transformations. | background 0.66 | 0.03 | 0.02 | asserted 1.00 |
| s6 | This last sort of transformation is available to us in the case of exactly triply regular association schemes. | method 0.66 | 0.20 | 0.03 | asserted 0.99 |
| s7 | The paper includes open problems and directions for future research. | future work 1.00 | 0.58 | 0.01 | asserted 0.96 |
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The paper
Original abstract
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 from three tools. A remarkable recent result of Man\v{c}inska and Roberson shows that graphs $G$ and $H$ are quantum isomorphic if and only if, for any planar graph $F$, the number of graph homomorphisms from $F$ to $G$ is equal to the number of graph homomorphisms from $F$ to $H$. A generalization of partition functions called "scaffolds" affords some basic reduction rules such as series-parallel reduction and can be applied to counting homomorphisms. The final tool is the classical theorem of Epifanov showing that any plane graph can be reduced to a single vertex and no edges by extended series-parallel reductions and Delta-Wye transformations. This last sort of transformation is available to us in the case of exactly triply regular association schemes. The paper includes open problems and directions for future research.
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How the literature uses it
2 arXiv papers cite this in 5 sentences
Papers citing it · 2 sentences without a confident label
Compares with it 1 paper
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As we mentioned in the introduction, this theorem was at the same time independently proved by [ref] .
Quantum symmetries of Hadamard matrices · Hadamard graphs · 0.96
Cites it as background 1 paper
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Recently, Chan and Martin [ref] and Gromada [ref] have shown that Hadamard graphs of the same order are quantum isomorphic.
Quantum isomorphic strongly regular graphs from the $E_8$ root system · Introduction · 0.81
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