Abstract
Graphs with large isoperimetric constants play an important role in cryptography because one can utilize such graphs to construct cryptographic hash functions. Ramanujan graphs are important optimal examples of such graphs, and known explicit construction of infinite families of Ramanujan graphs are given by Cayley graphs. A group–subgroup pair graph, which is a generalization of a Cayley graph, is defined for a given triplet consisting of finite group, its subgroup, and a suitable subset of the group. We study the spectra, that is the eigenvalues of the adjacency operators, of such graphs. In fact, we give an explicit formula of the eigenvalues of such graphs when the corresponding subgroups are abelian in terms of the characters of the subgroups as well as give a lower bound estimation for the second largest eigenvalues.
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References
C. Ballantine, D. Ciubotaru, Ramanujan bigraphs associated with \(SU(3)\) over a \(p\)-adic field. Proc. Amer. Math. Soc. 139, 1939–1953 (2011)
C. Ballantine, B. Feigon, R. Ganapathy, J. Kool, K. Maurischat, A. Wooding, Explicit construction of Ramanujan bigraphs, in Women in Numbers Europe: Research Directions in Number Theory, vol. 2, Association for Women in Mathematics Series, ed. by M.J. Bertin, et al. (Springer, Berlin, 2015), pp. 1–16
D. Charles, K. Lauter, E. Goren, Cryptographic hash functions from expander graphs. J. Cryptol. 22, 93–113 (2009)
K. Feng, W.-C.W. Li, Spectra of hypergraphs and applications. J. Number Theory 60(1), 1–22 (1996)
J. Friedman, J.-P. Tillich, Generalized alon-boppana theorems and error-correcting codes. SIAM J. Discret. Math. 19, 700–718 (2005)
K. Hashimoto, Zeta functions of finite graphs and representations of \(p\)-adic groups. Automorphic forms and geometry of arithmetic varieties, 211–280, Adv. Stud. Pure Math. 15. Academic Press, Boston (1989)
A. Lubotzky, R. Phillips, P. Sarnak, Ramanujan graphs. Combinatorica 8, 261–277 (1988)
B. Mohar, A strengthening and a multipartite generalization of the alon-boppana-Serre theorem. Proc. Amer. Math. Soc. 138, 3899–3909 (2010)
A.K. Pizer, Ramanujan graphs and Hecke operators. Bull. Am. Math. Soc. (N.S.) 23, 127–137 (1990)
C. Reyes-Bustos, Cayley-type graphs for group-subgroup pairs. Linear Algebr. Appl. 488, 320–349 (2016)
A. Sarveniazi, Explicit construction of a ramanujan \((n_1, n_2,\dots, n_{d-1})\)-regular hypergraph. Duke Math. J. 139, 141–171 (2007)
P. Solé, Ramanujan hypergraphs and Ramanujan geometries. Emerging applications of number theory (Minneapolis, MN, 1996), 583–590, IMA Vol. Math. Appl. 109 (Springer, New York, 1999)
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Kimoto, K. (2018). Spectra of Group-Subgroup Pair Graphs. In: Takagi, T., Wakayama, M., Tanaka, K., Kunihiro, N., Kimoto, K., Duong, D. (eds) Mathematical Modelling for Next-Generation Cryptography. Mathematics for Industry, vol 29. Springer, Singapore. https://doi.org/10.1007/978-981-10-5065-7_8
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DOI: https://doi.org/10.1007/978-981-10-5065-7_8
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