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Relationship Between Homogeneous Bent Functions and Nagy Graphs

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Abstract

We study the relationship between homogeneous bent functions and some intersection graphs of a special type that are called Nagy graphs and denoted by Γ(n,k). The graph Γ(n,k) is the graph whose vertices correspond to (nk) unordered subsets of size k of the set 1,..., n. Two vertices of Γ(n,k) are joined by an edge whenever the corresponding k-sets have exactly one common element. Those n and k for which the cliques of size k + 1 are maximal in Γ(n,k) are identified. We obtain a formula for the number of cliques of size k + 1 in Γ(n,k) for n = (k + 1)k/2. We prove that homogeneous Boolean functions of 10 and 28 variables obtained by taking the complement to the cliques of maximal size in Γ(10,4) and Γ(28,7) respectively are not bent functions.

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Funding

The author was supported by the Russian Foundation for Basic Research (project no. 18-07- 01394) and the Ministry of Science and Higher Education of the Russian Federation (Contract no. 1.13559.2019/13.1 and the Programme 5-100).

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Authors and Affiliations

  1. Novosibirsk State University, ul. Pirogova 2, Novosibirsk, 630090, Russia

    A. S. Shaporenko

  2. JetBrains Research, ul. Pirogova 1, Novosibirsk, 630090, Russia

    A. S. Shaporenko

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  1. A. S. Shaporenko
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Correspondence to A. S. Shaporenko.

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Russian Text © The Author(s), 2019, published in Diskretnyi Analiz i Issledovanie Operatsii, 2019, Vol. 26, No. 4, pp. 121–131.

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Shaporenko, A.S. Relationship Between Homogeneous Bent Functions and Nagy Graphs. J. Appl. Ind. Math. 13, 753–758 (2019). https://doi.org/10.1134/S1990478919040173

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  • DOI: https://doi.org/10.1134/S1990478919040173

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