High Quality Content by WIKIPEDIA articles! A Ramanujan graph, named after Srinivasa Ramanujan, is a regular graph whose spectral gap is almost as large as possible (see extremal graph theory). Such graphs are excellent spectral expanders. Examples of Ramanujan graphs include the clique, the biclique Kn,n, and the Petersen graph. As Murty's survey paper notes, Ramanujan graphs "fuse diverse branches of pure mathematics, namely, number theory, representation theory, and algebraic geometry". Constructions of Ramanujan graphs are often algebraic. Lubotzky, Phillips and Sarnak show how to construct an infinite family of p +1-regular Ramanujan graphs, whenever p ? 1 (mod 4) is a prime. Their proof uses the Ramanujan conjecture, which led to the name of Ramanujan graphs. Morgenstern extended the construction of Lubotzky, Phillips and Sarnak to all prime powers.

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