The nested split graphs whose second largest eigenvalue is equal to 1

Abstract

We determine all nested split graphs (NSG for short; i.e. graphs having no induced subgraphs equal to 2K2, P4, or C4) having the second largest eigenvalue equal to 1 and give some data regarding obtained results. The initial results in this research are given in the previous work of the second author, where all NSGs whose second largest eigenvalue is less than 1 are determined. It turns out that this case is well complicated with a number of solutions including some infinite families.