Dense triangle-free $(n, d, \lambda)$-graphs for all orders

Abstract

In 1994, Alon construct a triangle-free $(n,d,\lambda)$-graph with $d = \Omega(n^{2/3})$ and $\lambda = O(d^{1/2})$ for an exponentially increasing sequence of integers $n$. Using his ingenious construction, we deduce that there exist triangle-free $(n,d,\lambda)$-graphs with $d = \Omega(n^{2/3})$ and $\lambda = O( (d \log n)^{1/2} )$ for all sufficiently large $n$.