A proof of Alon's second eigenvalue conjecture

A d-regular graph has largest or first (adjacency matrix) eigenvalue 1 = d. In this paper we show the following conjecture of Alon. Fix an integer d > 2 and a real > 0. Then for sufficiently large n we have that "most" d-regular graphs on n vertices have all their eigenvalues except 1 = d bounded by 2 d - 1 + in absolute value. (Alon conjectured this only for 2, but our methods, being trace methods, also bound negative eigenvalues.)

• Type: Conference
• Year: 2003
• Where: STOC
• Authors: Joel Friedman
• Keywords: Absolute Value, Adjacency Matrix, Algorithms, STOC 2003