Pizer, A.: Ramanujan Graphs and Hecke Operators. Bull. AMS (New Series) 23, 127--137 (1990) Home/Search Document Not in Database Summary Related Articles Check |
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The Spectral Geometry of k-regular Graphs - Brooks (1991) (1 citation) (Correct)
....and see Theorem 4.3 below) this interval is asymptotically best possible for large graphs. Theorem 3. 2 then gives a necessary and sufficient condition for a graph to be Ramanujan, in terms of the asymptotics of N ( Gamma; We remark that in the construction of Ramanujan graphs in [19] and [20], it is the function N ( Gamma; which is readily accessible, rather than the eigenfunctions themselves. The function N ( Gamma; has, in these cases, a natural number theoretic interpretation, and the estimate ff = 1=2 follows from some rather deep results from number theory. We remark in ....
A. Pizer, "Ramanujan Graphs and Hecke Operators," Bull. AMS 23 (1990), pp. 127-137.
The Number-Theoretical Spin Chain and the Riemann Zeroes - Knauf (1998) (2 citations) (Correct)
....of the norm. 2 Ramanujan graphs are in a sense optimal, since the estimate c is always violated for large k regular graphs if c 2 p k Gamma 1. They have numbertheoretic and engineering applications. In the articles [3] of Chiu, 17] by Lubotzky, Phillips and Sarnak, 18] by Margulis, [22, 23] by Pizer and the books [24] by Sarnak and [16] by Lubotzky one finds different constructions leading to Ramanujan graphs. Ref. 26] by Venkov and Nikitin is a general survey. However, our family of graphs described in Def. 12 does not fall in one of these known families of Ramanujan graphs. 7 ....
Pizer, A.: Ramanujan Graphs and Hecke Operators. Bull. AMS (New Series) 23, 127--137 (1990)
The Number-Theoretical Spin Chain and the Riemann Zeroes - Knauf (1997) (2 citations) (Correct)
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Pizer, A.: Ramanujan Graphs and Hecke Operators. Bull. AMS (New Series) 23, 127--137 (1990)
Cryptographic Hash Functions from Expander Graphs - Charles, Goren, Lauter (2006) (Correct)
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Pizer, A.K.; Ramanujan Graphs and Hecke Operators, Bulletin of the AMS, Volume 23, Number 1, July 1990.
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