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

TL;DR

This paper characterizes simple connected graphs with second largest eigenvalue less than 1/2, identifying 13 specific classes and proposing bounds for the limit points of these eigenvalues.

Contribution

It provides a complete characterization of graphs with second largest eigenvalue below 1/2 and introduces bounds for the limit points of such eigenvalues.

Findings

13 classes of graphs with second largest eigenvalue < 1/2

Bounds for the limit points of second largest eigenvalues

Open problem regarding the exact value of c_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.