In spectral graph theory, 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 , 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". As an example of this, a regular graph is Ramanujan if and only if its Ihara zeta function satisfies an analog of the Riemann hypothesis.
Let be a connected -regular graph with vertices, and let be the eigenvalues of the adjacency matrix of . Because is connected and -regular, its eigenvalues satisfy . Whenever there exists with , define
A -regular graph is a Ramanujan graph if is defined and .
Extremality of Ramanujan graphs
Whenever is -regular and connected on at least three vertices, , and therefore . Let be the set of all connected -regular graphs with at least vertices. Because the minimum diameter of graphs in approaches infinity for fixed and increasing , Nilli's theorem implies an earlier theorem of Alon and Boppana which states
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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