We associate to the Hecke operator Tp , p a prime, acting on a space of theta series an explicit p + 1 regular Ramanujan graph G having large girth. Such graphs have high "magnification" and thus have many applications in the construction of networks and explicit algorithms (see [LPS1] and Bien's survey article [B]). In general our graphs do not seem to have quite as large a girth as the Ramanujan graphs discovered by Lubotzky, Phillips, and Sarnak ([LPS1, LPS3]) and independently by Margulis ([M]). However, by varying the T and the spaces of theta series, we obtain a much larger family of interesting graphs. The trace formula for the action of the Hecke operators Tpr immediately yields information on certain closed walks in G and in particular on the girth of G. If m is not a prime, we obtain "almost Ramanujan" graphs associated to Tm. The results of this paper can be viewed as an explicit version of a generalization of a construction of Ihara (see [I] and Theorem 4.1 of [LPS2]). From this viewpoint the connection between our results and those of Lubotzky, Phillips, and Sarnak becomes clearer. Recently, Chung ([C]) and Li ([L]) also constructed Ramanujan graphs associated to certain abelian groups.
[1]
Frank Harary,et al.
Graph Theory
,
2016
.
[2]
Martin Eichler,et al.
The Basis Problem for Modular Forms and the Traces of the Hecke Operators
,
1973
.
[3]
P. Deligne.
La conjecture de Weil. I
,
1974
.
[4]
H. Hijikata.
Explicit formula of the traces of Hecke operators for $\Gamma_{0}(N)$
,
1974
.
[5]
The representability of modular forms by theta series
,
1976
.
[6]
On the arithmetic of quaternion algebras II
,
1976
.
[7]
A. Pizer,et al.
An algorithm for computing modular forms on Γ0(N)
,
1980
.
[8]
Theta series and modular forms of level $p^2M$
,
1980
.
[9]
Alexander Lubotzky,et al.
Explicit expanders and the Ramanujan conjectures
,
1986,
STOC '86.
[10]
A. Lubotzky,et al.
Hecke operators and distributing points on S2. II
,
1987
.
[11]
F. Bien.
Constructions of telephone networks by group representations
,
1989
.
[12]
A. Pizer,et al.
The basis problem for modular forms on Γ
,
1989
.
[13]
Orders in quaternion algebras.
,
1989
.
[14]
F. Chung.
Diameters and eigenvalues
,
1989
.
[15]
J. Brzezinski.
On automorphisms of quaternion orders.
,
1990
.