Srinivasa Ramanujan
A regular graph whose spectral gap is almost as large as possible, indirectly named after Srinivasa Ramanujan via the Ramanujan–Petersson conjecture.
Srinivasa Ramanujan Iyengar was an Indian mathematician active in the early twentieth century who produced profound contributions to mathematical analysis, number theory, infinite series, and continued fractions. He famously solved problems that had been considered unsolvable. Initially working in isolation, Ramanujan developed his own research independently. According to psychologist Hans Eysenck, Ramanujan attempted to interest leading professional mathematicians in his work but largely failed, as his ideas were too novel, unfamiliar, and presented in unusual ways for them to engage with. In 1913, seeking mathematicians who could better appreciate his findings, he began corresponding by mail with the English mathematician G. H. Hardy at the University of Cambridge. Hardy recognized Ramanujan’s work as extraordinary and arranged for him to travel to Cambridge. Hardy noted that Ramanujan had produced groundbreaking new theorems, some of which completely defeated him, and others that were highly advanced results only recently proven. Over his lifetime, Ramanujan independently compiled nearly 3,900 results, mostly identities and equations. Many were entirely novel, including the Ramanujan prime, the Ramanujan theta function, partition formulae, and mock theta functions, which opened entire new areas of research. The vast majority of his thousands of results have since been proven correct. The Ramanujan Journal was established to publish work influenced by his ideas, and his notebooks have been studied for decades as sources of new mathematical concepts. As late as 2012, researchers discovered that mere comments in his writings about "simple properties" were themselves profound number theory results unsuspected for nearly a century. He became one of the youngest Fellows of the Royal Society and only the second Indian member, as well as the first Indian elected a Fellow of Trinity College, Cambridge. In 1919, ill health—now believed to be hepatic amoebiasis from earlier dysentery—forced his return to India, where he died in 1920 at age 32. His last letters to Hardy, from January 1920, show he continued producing new ideas. His "lost notebook," containing discoveries from his final year, caused great excitement when rediscovered in 1976.
- field
- Mathematics
Lore & Background
The complete graph K_{d+1} has spectrum d, -1, -1, …, -1, and thus λ(K_{d+1}) = 1 and the graph is a Ramanujan graph for every d > 1. The complete bipartite graph K_{d,d} has spectrum d, 0, 0, …, 0, -d and hence is a bipartite Ramanujan graph for every d. The Petersen graph has spectrum 3, 1, 1, 1, 1, 1, -2, -2, -2, -2, so it is a 3-regular Ramanujan graph. The icosahedral graph is a 5-regular Ramanujan graph. A Paley graph of order q is (q-1)/2-regular with all other eigenvalues being (-1 ± √q)/2, making Paley graphs an infinite family of Ramanujan graphs. Lubotzky, Phillips and Sarnak and independently Margulis showed how to construct an infinite family of (p+1)-regular Ramanujan graphs, whenever p is a prime number and p ≡ 1 (mod 4). Both proofs use the Ramanujan conjecture, which led to the name of Ramanujan graphs.
Reader's Guide
A connected d-regular graph G is a Ramanujan graph if λ(G) ≤ 2√(d-1), where λ(G) = max_{i≠1} |λ_i|. Many sources use an alternative definition λ'(G) = max_{|λ_i|<d} |λ_i| to define Ramanujan graphs, allowing -d in addition to the "small" eigenvalues. Since λ_n = -d if and only if the graph is bipartite, graphs that satisfy this alternative definition but not the first definition are called bipartite Ramanujan graphs. If G is a Ramanujan graph, then G × K_2 is a bipartite Ramanujan graph, so the existence of Ramanujan graphs is stronger. As observed by Toshikazu Sunada, a regular graph is Ramanujan if and only if its Ihara zeta function satisfies an analog of the Riemann hypothesis. More generally, let f(x) be a degree 2 or 3 polynomial over F_q. Let S = {f(x) : x ∈ F_q} be the image of f(x) as a multiset, and suppose S = -S. Then the Cayley graph for F_q with generators from S is a Ramanujan graph.
Did You Know?
- The name 'Ramanujan graph' comes from the Ramanujan–Petersson conjecture, which was used in a construction of some of these graphs.
- The Petersen graph is a 3-regular Ramanujan graph with spectrum 3, 1, 1, 1, 1, 1, -2, -2, -2, -2.
- A Paley graph of order q is an infinite family of Ramanujan graphs.
- Lubotzky, Phillips and Sarnak and independently Margulis constructed infinite families of (p+1)-regular Ramanujan graphs for primes p ≡ 1 (mod 4).
- As observed by Toshikazu Sunada, a regular graph is Ramanujan if and only if its Ihara zeta function satisfies an analog of the Riemann hypothesis.
Frequently Asked Questions
Who is Srinivasa Ramanujan?
Srinivasa Ramanujan Iyengar was an Indian mathematician born on 22 December 1887 who produced an extraordinary volume of theorems largely through self-directed study. He is credited with nearly 3,900 results across multiple branches of mathematics.
What are Srinivasa Ramanujan's powers/role?
Ramanujan's core contributions lie in mathematical analysis, number theory, infinite series, and continued fractions. He is especially remembered for the Ramanujan prime, the Ramanujan theta function, partition formulae, and mock theta functions.
How does Srinivasa Ramanujan's story end?
Ramanujan passed away on 26 April 1920 at the age of 32 while living in Cambridge, England. His death cut short a career that had barely begun to reach its full potential after Hardy brought him to Cambridge.
Why is Srinivasa Ramanujan important?
He solved problems that were widely regarded as unsolvable and produced results of remarkable depth without any formal mathematical training. His work in partition theory, mock theta functions, and related areas remains an active area of research more than a century later.
How did Ramanujan connect with G. H. Hardy?
Ramanujan developed his research entirely in isolation before reaching out to the English mathematician G. H. Hardy. Hardy recognized the work as extraordinary and arranged for Ramanujan to travel to Cambridge, where their partnership took shape.
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