Graphs with second largest eigenvalue less than $1/2$

Abstract

We characterize the simple connected graphs with the second largest eigenvalue less than 1/2, which consists of 13 classes of specific graphs. These 13 classes hint that $c_{2}\in [1/2, \sqrt{2+\sqrt{5}}]$, where $c_2$ is the minimum real number $c$ for which every real number greater than $c$ is a limit point in the set of the second largest eigenvalues of the simple connected graphs. We leave it as a problem.