Producing the graphs of Lubotzky Phillips and Sarnak in Matlab

Ramanujan graphs

Generalized Triangle Groups, Expanders, and a Problem of Agol and Wise

Answering a question asked by Agol and Wise, we show that a desired stronger form of Wise’s malnormal special quotient theorem does not hold. The counterexamples are generalizations of triangle groups, built using the Ramanujan graphs constructed by Lubotzky–Phillips–Sarnak.

Expanding graphs, Ramanujan graphs, and 1-factor perturbations

We construct (k±1)-regular graphs which provide sequences of expanders by adding or substracting appropriate 1-factors from given sequences of kregular graphs. We compute numerical examples in a few cases for which the given sequences are from the work of Lubotzky, Phillips, and Sarnak (with k − 1 the order of a finite field). If k + 1 = 7, our construction results in a sequence of 7-regular ex...

A New Series of Dense Graphs of High Girth

Let k ≥ 1 be an odd integer, t = b k+2 4 c, and q be a prime power. We construct a bipartite, q–regular, edge–transitive graph CD(k, q) of order v ≤ 2qk−t+1 and girth g ≥ k + 5. If e is the the number of edges of CD(k, q), then e = Ω(v 1 k−t+1 ). These graphs provide the best known asymptotic lower bound for the greatest number of edges in graphs of order v and girth at least g, g ≥ 5, g 6= 11,...

Families of Ramanujan Graphs and Quaternion Algebras

Expander graphs are graphs in which the neighbors of any given “not too large” set of vertices X form a large set relative to the size of X—rumors tend to spread very fast. Among those, the Ramanujan graphs are extremal in their expansion properties. To be precise, the eigenvalues of the adjacency matrix have an extremal property that guarantees good expansion properties. Expanders, and hence R...