Ramanujan Graphs (1711.06558v1)

Abstract: This is an item on Ramanujan Graphs for a planned encyclopedia on Ramanujan. The notion of Ramanujan graphs is explained, as well as the reason to name these graphs after Ramanujan.

Insights on "Ramanujan Graphs" by Alexander Lubotzky

Alexander Lubotzky's paper on Ramanujan graphs delivers an incisive examination of a class of graphs that have become integral to combinatorics and theoretical computer science. These graphs, which satisfy certain spectral conditions, optimize their expansion properties, making them invaluable in various domains such as network design and algorithm efficiency. The paper not only defines Ramanujan graphs but also elucidates their intrinsic connections to deep results in algebraic number theory.

Definition and Spectral Properties

A k-regular graph X with n vertices is termed a Ramanujan graph if its nontrivial eigenvalues are bounded by $|λ| \leq 2\sqrt{k-1}$. This is notable as it represents the spectral bound dictated by the Alon-Boppana theorem, making these graphs optimal expanders. The fundamental underpinning showcased in the paper is the circumstantial significance of their eigenvalue distribution, which aligns with the conjectures posited by Ramanujan, albeit originating in a different mathematical discipline.

Expander Graphs and Applications

Ramanujan graphs are distinguished as excellent expanders. Critical in combinatorial constructions and having important implications in the design of efficient algorithms, expander graphs facilitate rapid mixing of random walks, delivering towards more uniform distributions. This property, particularly pronounced in Ramanujan graphs due to their minimized eigenvalue gaps, is vital for applications in randomness extraction and error-correcting codes.

Construction and Existence

While random k-regular graphs are known extenders, the determination of their Ramanujan status remains unresolved. The paper references explicit constructions by Lubotzky, Phillips, and Sarnak (LPS), demonstrating the existence of infinite families of Ramanujan graphs for $k = q + 1$, $q$ a prime. Later, the existence was extended non-constructively by Marcus, Spielman, and Srivastava for all $k \geq 3$.

Theoretical Foundations and Connections

Lubotzky elaborates on the theoretical underpinning relating to the Ramanujan conjecture through modular forms and representation theory. The Ramanujan-Peterson conjecture, extended to the spectral properties of operators on automorphic forms, finds striking parallels in the eigenvalue constraints of these graphs. Deligne's proof of this conjecture for specific representations, paired with the Jacquet-Langlands correspondence, facilitates the construction of Ramanujan graphs via arithmetic groups.

Moreover, the paper acknowledges the connection between Ramanujan graphs and the Ihara zeta function of a graph, which much like the Riemann hypothesis for number fields, underpins the notion that the absence of eigenvalues outside the critical spectrum ensures Ramanujan properties.

Implications and Future Developments

The insights delivered by Lubotzky not only reinforce the utility of Ramanujan graphs in current mathematical applications but also open avenues for further generalizations in higher-dimensional analogs known as Ramanujan complexes. The deep interrelations with number theory, particularly through automorphic perspectives and spectral graph theory, indicate a fertile ground for further exploration and potential advancements in computational paradigms.

In summation, Alexander Lubotzky's exposition on Ramanujan graphs elucidates a profound synthesis of graph theory and algebraic number theory, positioning these graphs as essential tools in both theoretical investigations and practical applications within computer science and combinatorial design.