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Ramanujan graphs. (English) Zbl 0914.05051
Buell, D. A. (ed.) et al., Computational perspectives on number theory. Proceedings of a conference in honor of A. O. L. Atkin, Chicago, IL, USA, September 1995. Providence, RI: American Mathematical Society. AMS/IP Stud. Adv. Math. 7, 159-178 (1998).
A connected regular graph of degree $k$ is said to be a Ramanujan graph if all its eigenvalues (other than $±k$) have modulus at most $2\sqrt{k-1}$. The author constructs some new Ramanujan graphs using the arithmetic of quaternion algebras, Hecke operators and modular forms.
##### MSC:
 05C50 Graphs and linear algebra 05C25 Graphs and abstract algebra
##### Keywords:
Ramanujan graph; eigenvalues