Ramanujan graphs and expander families constructed from p-ary bent functions
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We present a method for constructing an infinite family of non-bipartite Ramanujan graphs. We mainly employ p-ary bent functions of \((p-1)\)-form for this construction, where p is a prime number. Our result leads to construction of infinite families of expander graphs; this is due to the fact that Ramanujan graphs play as base expanders for constructing further expanders. For our construction we directly compute the eigenvalues of the Ramanujan graphs arsing from p-ary bent functions. Furthermore, we establish a criterion on the regularity of p-ary bent functions in m variables of \((p-1)\)-form when m is even. Finally, using weakly regular p-ary bent functions of \(\ell \)-form, we find (amorphic) association schemes in a constructive way; this resolves the open case that \(\ell = p-1\) for \(p >2\) for finding (amorphic) association schemes.
KeywordsRamanujan graph p-ary bent function Expanders (amorphic)association scheme
Mathematics Subject ClassificationPrimary 05E30 Secondary 11T71 94C10
J.Y. Hyun was supported by the National Research Foundation of Korea (NRF) grant funded by the Korea government (MEST) (NRF-2017R1D1A1B05030707), J. Lee by the National Research Foundation of Korea (NRF) grant funded by the Korea government (MEST) (NRF-2017R1A6A3A11030486) and 2019 Research Grant from Kangwon National University, and Y. Lee by Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education (Grant No. 2019R1A6A1A11051177) and also by the National Research Foundation of Korea (NRF) grant funded by the Korea government (MEST) (NRF-2017R1A2B2004574). We express our gratitude to the reviewers for their very helpful comments, which lead to improvement of the exposition of this paper.
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