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In Search of Hyperpaths

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Abstract

Hypertrees are high-dimensional counterparts of graph theoretic trees. They have attracted a great deal of attention by various investigators. Here we introduce and study hyperpaths—a particular class of hypertrees which are high dimensional analogs of paths in graph theory. A d-dimensional hyperpath is a d-dimensional hypertree in which every \((d-1)\)-dimensional face is contained in at most \((d+1)\) faces of dimension d. We introduce a possibly infinite family of hyperpaths for every dimension, and investigate its properties in greater depth for dimension \(d=2\).

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Notes

  1. Throughout this paper, unless stated otherwise, given a prime n, all arithmetic equations are mod n, and we often replace the congruence relation \(\equiv \) by an equality sign when no confusion is possible.

References

  1. Aronshtam, L., Linial, N., Łuczak, T., Meshulam, R.: Collapsibility and vanishing of top homology in random simplicial complexes. Discrete Comput. Geom. 49(2), 317–334 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  2. Bläser, M.: Fast matrix multiplication and related problems. http://www.ens-lyon.fr/LIP/MC2/data/uploads/slidesmarkusblaser.pdf

  3. Dahari, A., Linial, N.: Hyperpaths (2020). arXiv:2011.09936

  4. Davis, P.J.: Circulant Matrices. A Wiley-Interscience Publication. Pure and Applied Mathematics. Wiley, New York (1979)

    Google Scholar 

  5. Gauss, C.F.: Untersuchungen über höhere Arithmetik. American Mathematical Society, Providence (2006)

    Google Scholar 

  6. Hardy, G.H., Wright, E.M.: An Introduction to the Theory of Numbers. Oxford University Press, New York (1979)

    MATH  Google Scholar 

  7. Kalai, G.: Enumeration of Q-acyclic simplicial complexes. Israel J. Math. 45(4), 337–351 (1983)

    Article  MathSciNet  MATH  Google Scholar 

  8. Le Gall, F.: Powers of tensors and fast matrix multiplication. In: 39th International Symposium on Symbolic and Algebraic Computation (Kobe 2014), pp. 296–303. ACM, New York (2014)

  9. Linial, N., Meshulam, R., Rosenthal, M.: Sum complexes–a new family of hypertrees. Discrete Comput. Geom. 44(3), 622–636 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  10. Linial, N., Newman, I., Peled, Y., Rabinovich, Y.: Extremal hypercuts and shadows of simplicial complexes. Israel J. Math. 229(1), 133–163 (2019)

    Article  MathSciNet  MATH  Google Scholar 

  11. Linial, N., Peled, Y.: On the phase transition in random simplicial complexes. Ann. Math. 184(3), 745–773 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  12. Linial, N., Peled, Y.: Enumeration and randomized constructions of hypertrees. Random Struct. Algorithms 55(3), 677–695 (2019)

    Article  MathSciNet  MATH  Google Scholar 

  13. Mathew, R., Newman, I., Rabinovich, Y., Rajendraprasad, D.: Hamiltonian and pseudo-Hamiltonian cycles and fillings in simplicial complexes. J. Combin. Theory Ser. B 150, 119–143 (2021)

    Article  MathSciNet  MATH  Google Scholar 

  14. McDonald, B.R.: Linear Algebra Over Commutative Rings. Monographs and Textbooks in Pure and Applied Mathematics, vol. 87. Marcel Dekker, New York (1984)

  15. Petković, M.D., Stanimirović, P.S.: Generalized matrix inversion is not harder than matrix multiplication. J. Comput. Appl. Math. 230(1), 270–282 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  16. Rjasanow, S.: Effective algorithms with circulant-block matrices. Linear Algebra Appl. 202, 55–69 (1994)

    Article  MathSciNet  MATH  Google Scholar 

  17. Tsitsas, N.L., Alivizatos, E.G., Kalogeropoulos, G.H.: A recursive algorithm for the inversion of matrices with circulant blocks. Appl. Math. Comput. 188(1), 877–894 (2007)

    MathSciNet  MATH  Google Scholar 

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Acknowledgements

We thank Roy Meshulam for insightful comments on this manuscript.

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Authors and Affiliations

  1. School of Computer Science and Engineering, Hebrew University, 91904, Jerusalem, Israel

    Amir Dahari & Nathan Linial

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  1. Amir Dahari
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  2. Nathan Linial
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Correspondence to Amir Dahari.

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Supported by BSF US-Israel Grant 2018313 “Between topology and combinatorics”

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Dahari, A., Linial, N. In Search of Hyperpaths. Discrete Comput Geom 69, 399–421 (2023). https://doi.org/10.1007/s00454-021-00360-x

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  • DOI: https://doi.org/10.1007/s00454-021-00360-x

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