Sarnak and independently Margulis showed how to construct an infinite family of -regular Ramanujan graphs, whenever is a prime number and . Both proofs use the Ramanujan conjecture, which led to the name of Ramanujan graphs. Besides being Ramanujan graphs, these constructions satisfies some other properties, for example, their girth is where is the number of nodes. Let us sketch the Lubotzky-Phillips-Sarnak construction. Let be a prime not equal to . By Jacobi's four-square theorem, there are solutions to the equation where is odd and are even. To each such solution associate the matrix If is a quadratic residue modulo let be the Cayley graph of with these generators, and otherwise, let be the Cayley graph of with the same generators. Then is a -regular graph on or vertices depending on whether or not is a quadratic residue modulo . It is proved that is a Ramanujan graph. Morgenstern later extended the construction of Lubotzky, Phillips and Sarnak. His extended construction holds whenever is a prime power. Arnold Pizer proved that the supersingular isogeny graphs are Ramanujan, although they tend to have lower girth than the graphs of Lubotzky, Phillips, and Sarnak. Like the graphs of Lubotzky, Phillips, and Sarnak, the degrees of these graphs are always a prime number plus one. Probabilistic examples Adam Marcus, Daniel Spielman and Nikhil Srivastava proved the existence of infinitely many -regular bipartite Ramanujan graphs for any . Later they proved that there exist bipartite Ramanujan graphs of every degree and every number of vertices. Michael B. Cohen showed how to construct these graphs in polynomial time. The initial work followed an approach of Bilu and Linial. They considered an operation called a 2-lift that takes a -regular graph with vertices and a sign on each edge, and produces a new -regular graph on vertices. Bilu & Linial conjectured that there always exists a signing so that every new eigenvalue of has magnitude at most . This conjecture guarantees the existence of Ramanujan graphs with degree and vertices for any —simply start with the complete graph , and iteratively take 2-lifts that retain the Ramanujan property. Using the method of interlacing polynomials, Marcus, Spielman, and Srivastava proved Bilu & Linial's conjecture holds when is already a bipartite Ramanujan graph, which is enough to conclude the existence result. The sequel proved the stronger statement that a sum of random bipartite matchings is Ramanujan with non-vanishing