Abstract
We relate the one- and two-variable interlace polynomials of a graph to the spectra of a quadratic boolean function with respect to a strategic subset of local unitary transforms. By so doing we establish links between graph theory, cryptography, coding theory, and quantum entanglement. We establish the form of the interlace polynomial for certain functions, provide new one and two-variable interlace polynomials, and propose a generalisation of the interlace polynomial to hypergraphs. We also prove conjectures from [15] and equate certain spectral metrics with various evaluations of the interlace polynomial.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
Aigner, M., van der Holst, H.: Interlace Polynomials. Linear Algebra and its Applications 377, 11–30 (2004)
Arratia, R., Bollobas, B., Sorkin, G.B.: The Interlace Polynomial: a new graph polynomial. In: Proc. 11th Annual ACM-SIAM Symp. on Discrete Math., pp. 237–245 (2000)
Arratia, R., Bollobas, B., Sorkin, G.B.: The Interlace Polynomial of a Graph. J. Combin. Theory Ser. B 92(2), 199–233 (2004), http://arxiv.org/pdf/math/0209045 (v2 August 13, 2004)
Arratia, R., Bollobas, B., Sorkin, G.B.: Two-Variable Interlace Polynomial. Combinatorica 24(4), 567–584 (2004), http://arxiv.org/pdf/math/0209054 (v3 August 13, 2004)
Balister, P.N., Bollobas, B., Cutler, J., Pebody, L.: The Interlace Polynomial of Graphs at − 1. Europ. J. Combinatorics 23, 761–767 (2002)
Bouchet, A.: Tutte-Martin Polynomials and Orienting Vectors of Isotropic Systems. Graphs Combin. 7, 235–252 (1991)
Calderbank, A.R., Rains, E.M., Shor, P.W., Sloane, N.J.A.: Quantum Error Correction Via Codes Over GF(4). IEEE Trans. on Inform. Theory 44, 1369–1387 (1998), http://xxx.soton.ac.uk/pdf/quant-ph/?9608006
Danielsen, L.E.: On Self-Dual Quantum Codes, Graphs, and Boolean Functions. Master’s Thesis, Selmer Centre, Inst. for Informatics, University of Bergen, Bergen, Norway (March 2005), http://arxiv.org/pdf/quant-ph/0503236
Danielsen, L.E., Gulliver, T.A., Parker, M.G.: Aperiodic Propagation Criteria for Boolean Functions. Inform. Comput. (accepted, 2005), http://www.ii.uib.no/~matthew/apcpaper.pdf
Danielsen, L.E., Parker, M.G.: Spectral Orbits and Peak-to-Average Power Ratio of Boolean Functions with Respect to the {I,H,N}n Transform. In: Helleseth, T., Sarwate, D., Song, H.-Y., Yang, K. (eds.) SETA 2004. LNCS, vol. 3486, pp. 373–388. Springer, Heidelberg (2005), http://xxx.soton.ac.uk/pdf/cs.IT/0504102
Glynn, D.G.: On Self-Dual Quantum Codes and Graphs. Electronic Journal of Combinatorics (April 2002) (submitted), http://homepage.mac.com/dglynn/quantum_files/Personal3.html
Gulliver, T.A., Parker, M.G.: The Multivariate Merit Factor of a Boolean Function. In: IEEE ITSOC Information Theory Workshop 2005 on Coding and Complexity, Rotorua, New Zealand, August 29 - September 1 (2005), http://www.ii.uib.no/~matthew/NZRecursionsCamera1.pdf
Hein, M., Eisert, J., Briegel, H.J.: Multi-Party Entanglement in Graph States. Phys. Rev. A 69(6) (2004), http://xxx.soton.ac.uk/pdf/quant-ph/0307130
MacWilliams, F.J., Sloane, N.J.A.: The Theory of Error-Correcting Codes. North-Holland, Amsterdam (1977)
Parker, M.G.: Constabent Properties of Golay-Davis-Jedwab Sequences. In: ISIT2000, Sorrento, Italy (June 2000), http://www.ii.uib.no/~matthew/BentGolayISIT.ps
Parker, M.G.: Generalised S-Box Nonlinearity. NESSIE Public Document - NES/DOC/UIB/WP5/020/A, February 11 (2003), https://www.cosic.esat.kuleuven.ac.be/nessie/reports/phase2/SBoxLin.pdf
Parker, M.G.: Univariate and Multivariate Merit Factors. In: Helleseth, T., Sarwate, D., Song, H.-Y., Yang, K. (eds.) SETA 2004. LNCS, vol. 3486, pp. 72–100. Springer, Heidelberg (2005), http://www.ii.uib.no/~matthew/seta04-mf.pdf
Parker, M.G., Rijmen, V.: The Quantum Entanglement of Binary and Bipolar Sequences. In: Sequences and Their Applications. Discrete Mathematics and Theoretical Computer Science Series. Springer, Heidelberg (2001), http://xxx.soton.ac.uk/ps/quant-ph/?0107106 , http://www.ii.uib.no/~matthew/BergDM3.ps
M. G. Parker and C. Tellambura, ”A Construction for Binary Sequence Sets with Low Peak-to-Average Power Ratio”, Technical Report No 242, Dept. of Informatics, University of Bergen, Norway, http://www.ii.uib.no/ publikasjoner/texrap/ps/2003-242.ps, Feb. 2003.
Riera, C., Parker, M.G.: Generalised Bent Criteria for Boolean Functions. IEEE Trans Inform. Theory (July 2005) (accepted), http://xxx.soton.ac.uk/pdf/cs.IT/0502049
Riera, C., Petrides, G., Parker, M.G.: Generalised Bent Criteria for Boolean Functions (II), http://xxx.soton.ac.uk/pdf/cs.IT/0502050
Riera, C., Parker, M.G.: On Pivot Orbits of Boolean Functions. In: Proceedings of the Fourth International Workshop on Optimal Codes and Related Topics (OC 2005), Pamporovo, Bulgaria (June 2005), http://www.ii.uib.no/~matthew/octalk4.ps
Rothaus, O.S.: On Bent Functions. J. Comb. Theory 20A, 300–305 (1976)
Wei, T., Ericsson, M., Goldbart, P.M., Munro, W.J.: Connections between relative entropy of entanglement and geometric measure of entanglement. Quantum Information and Computation 4, 252–272 (2004), http://xxx.soton.ac.uk/pdf/quant-ph/0405002
Wei, T., Goldbart, P.M.: Geometric measure of entanglement and applications to bipartite and multipartite quantum states. Physical Review A 68 (2003), http://xxx.soton.ac.uk/pdf/quant-ph/0307219
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2006 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Riera, C., Parker, M.G. (2006). One and Two-Variable Interlace Polynomials: A Spectral Interpretation. In: Ytrehus, Ø. (eds) Coding and Cryptography. WCC 2005. Lecture Notes in Computer Science, vol 3969. Springer, Berlin, Heidelberg. https://doi.org/10.1007/11779360_31
Download citation
DOI: https://doi.org/10.1007/11779360_31
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-35481-9
Online ISBN: 978-3-540-35482-6
eBook Packages: Computer ScienceComputer Science (R0)