Graphs with second largest eigenvalue less than 1/2

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) is an element of [1/2, root 2 + root 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. (c) 2023 Elsevier Inc. All rights reserved.

Automated summary

In this article, we characterize the simple connected graphs with the second largest eigenvalue less than 1/2 and identify 13 specific classes of graphs. These 13 classes imply that c(2) belongs to [1/2, root 2 + root 5], where c(2) is the minimum real number 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 an open problem.