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

Markström, Klas

doi:10.1201/9781315367996-7

Cited by 4 publications

(6 citation statements)

References 17 publications

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“…Relative to X G , only a few examples of nonisomorphic graphs with equal Tutte symmetric functions are known. An example with a minimum number of vertices and edges is given by Markstrom [30] 3 . Additionally, Brylawski [7] uses the rotor-like graph given in Figure 4 to construct a family of graph pairs with arbitrarily high connectivity and equal Tutte symmetric function 4 .…”

Section: Graphs With Equal Xbmentioning

confidence: 99%

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A Vertex-Weighted Tutte Symmetric Function, and Constructing Graphs with Equal Chromatic Symmetric Function

Aliste-Prieto

Crew

Spirkl

et al. 2021

Electron. J. Combin.

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This paper has two main parts. First, we consider the Tutte symmetric function XB, a generalization of the chromatic symmetric function. We introduce a vertex-weighted version of XB and show that this function admits a deletion-contraction relation. We also demonstrate that the vertex-weighted XB admits spanning-tree and spanning-forest expansions generalizing those of the Tutte polynomial by connecting XB to other graph functions. Second, we give several methods for constructing nonisomorphic graphs with equal chromatic and Tutte symmetric functions, and use them to provide specific examples.

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Section: Graphs With Equal Xbmentioning

confidence: 99%

“…The example of[30] is also Example 259 of the authors' list of pairs of graphs with equal chromatic symmetric function[11] 4. This construction depends in part on the fact that the Tutte symmetric function of a simple graph G uniquely determines that of G's complement.…”

mentioning

confidence: 99%

A Vertex-Weighted Tutte Symmetric Function, and Constructing Graphs with Equal Chromatic Symmetric Function

Aliste-Prieto

Crew

Spirkl

et al. 2021

Electron. J. Combin.

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“…[35], where they were found via a complete search of all small graphs, and the search was extended in Ref. [37], where non-isomorphic graphs which have the same partition function for all two-state spin models were found. But due to the rapidly growing number of non-isomorphic graphs, even when restricted to e.g.…”

Section: Appendix: Choice Of Graphsmentioning

confidence: 99%

“…Using data from Refs. [35,37] we use a simple version of this method to construct larger graphs to use as benchmarks.…”

Section: Appendix: Choice Of Graphsmentioning

confidence: 99%

Discriminating nonisomorphic graphs with an experimental quantum annealer

et al. 2020

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We demonstrate experimentally the ability of a quantum annealer to distinguish between sets of non-isomorphic graphs that share the same classical Ising spectrum. Utilizing the pause-and-quench features recently introduced into D-Wave quantum annealing processors, which allow the user to probe the quantum Hamiltonian realized in the middle of an anneal, we show that obtaining thermal averages of diagonal observables of 'classically indistinguishable' non-isomorphic graphs encoded into transverse-field Ising Hamiltonians enable their discrimination. We discuss the significance of our results in the context of the graph isomorphism problem.

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“…In fact, characterizing what kinds of graphs are determined by the per-spectra is generally a very hard problem. Up to now, only a few types of graphs are proved to be DPS; see [12][13][14][15][16][17].…”

Section: Introductionmentioning

confidence: 99%

Per-Spectral Characterizations of Bicyclic Networks

Lü

2017

Journal of Applied Mathematics

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Spectral techniques are used for the study of several network properties: community detection, bipartition, clustering, design of highly synchronizable networks, and so forth. In this paper, we investigate which kinds of bicyclic networks are determined by their per-spectra. We find that the permanental spectra cannot determine sandglass graphs in general. When we restrict our consideration to connected graphs or quadrangle-free graphs, sandglass graphs are determined by their permanental spectra. Furthermore, we construct countless pairs of per-cospectra bicyclic networks.

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

Cited by 4 publications

References 17 publications

A Vertex-Weighted Tutte Symmetric Function, and Constructing Graphs with Equal Chromatic Symmetric Function

A Vertex-Weighted Tutte Symmetric Function, and Constructing Graphs with Equal Chromatic Symmetric Function

Discriminating nonisomorphic graphs with an experimental quantum annealer

Per-Spectral Characterizations of Bicyclic Networks

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