Distinguishing graphs by their left and right homomorphism profiles

Garijo, Delia; Goodall, Andrew; Nešetřil, Jaroslav

doi:10.1016/j.ejc.2011.03.012

Cited by 15 publications

(14 citation statements)

References 23 publications

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“…This is denoted by . It is not hard to find graphs with the same Tutte polynomial (T-equivalent) that are not isomorphic 20 21 : for example, all trees on the same number of vertices.…”

Section: Resultsmentioning

confidence: 99%

Hearing the Shape of the Ising Model with a Programmable Superconducting-Flux Annealer

Vinci

Markström

Boixo

et al. 2014

Sci Rep

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Two objects can be distinguished if they have different measurable properties. Thus, distinguishability depends on the Physics of the objects. In considering graphs, we revisit the Ising model as a framework to define physically meaningful spectral invariants. In this context, we introduce a family of refinements of the classical spectrum and consider the quantum partition function. We demonstrate that the energy spectrum of the quantum Ising Hamiltonian is a stronger invariant than the classical one without refinements. For the purpose of implementing the related physical systems, we perform experiments on a programmable annealer with superconducting flux technology. Departing from the paradigm of adiabatic computation, we take advantage of a noisy evolution of the device to generate statistics of low energy states. The graphs considered in the experiments have the same classical partition functions, but different quantum spectra. The data obtained from the annealer distinguish non-isomorphic graphs via information contained in the classical refinements of the functions but not via the differences in the quantum spectra.

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“…This is denoted by . It is not hard to find graphs with the same Tutte polynomial (T-equivalent) that are not isomorphic 20 21 : for example, all trees on the same number of vertices.…”

Section: Resultsmentioning

confidence: 99%

Hearing the Shape of the Ising Model with a Programmable Superconducting-Flux Annealer

Vinci

Markström

Boixo

et al. 2014

Sci Rep

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“…Many thanks to Prof. A. Goodall (Charles University, Prague) and Prof. K. Markström (Umea University, Sweden) for sending me references [8] and [1,19], respectively. Special thanks to Prof. Asteroide Santana (UFSC) for help with latex commands.…”

Section: Acknowledgmentsmentioning

confidence: 99%

The Feynman Identity for Planar Graphs

Costa

2016

Lett Math Phys

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The Feynman identity (FI) of a planar graph relates the Euler polynomial of the graph to an infinite product over the equivalence classes of closed nonperiodic signed cycles in the graph. The main objectives of this paper are to compute the number of equivalence classes of nonperiodic cycles of given length and sign in a planar graph and to interpret the data encoded by the FI in the context of free Lie superalgebras. This solves in the case of planar graphs a problem first raised by S. Sherman and sets the FI as the denominator identity of a free Lie superalgebra generated from a graph. Other results are obtained. For instance, in connection with zeta functions of graphs.Mathematics Subject Classification. 82B20, 05A19, 17B01.

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“…One question which has been studied in connection with several of the graph polynomials already mentioned is whether non-isomorphic graphs can have the same polynomial, so far the answer has been yes for all polynomials studied, and how finely the given polynomial partitions the set of all graphs into equivalence classes. For the bivariate Ising polynomial this was done in [AM09], where arbitrarily large sets of non-isomorphic graphs with the same bivariate Ising polynomial were constructed, and in [GGN11] this was done for the Potts model partition function. This is a well studied question for the chromatic polynomial, see e.g.…”

Section: Non-isomorphic Graphs With the Same Homomorphism Polynomialmentioning

confidence: 99%

“…If some of the variables in P q (G) are given either numerical values or given equal weights this type of monotonicity is not necessarily preserved. In [GGN11] the authors asked for examples of graphs with the same q-state Potts model partition function for q = 3 but different ones for q = 2. We used the same computer programs as for our earlier classification to look for such examples as well.…”

Section: Non-isomorphic Graphs With the Same Homomorphism Polynomialmentioning

confidence: 99%

From the Ising and Potts Models to the General Graph Homomorphism Polynomial

Markström¹

2016

Graph Polynomials

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Distinguishing graphs by their left and right homomorphism profiles

Cited by 15 publications

References 23 publications

Hearing the Shape of the Ising Model with a Programmable Superconducting-Flux Annealer

Hearing the Shape of the Ising Model with a Programmable Superconducting-Flux Annealer

The Feynman Identity for Planar Graphs

From the Ising and Potts Models to the General Graph Homomorphism Polynomial

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