Connected claw-free graphs whose second largest eigenvalue does not exceed 1

Abstract

By extending the main result of [European J. Combin., 97 (2021), 103385], we determine all connected claw-free graphs whose second largest eigenvalue does not exceed 1, where a claw is the star K<inf>1,3</inf>. In contrast to the existing results on graphs with the same spectral property, our approach introduces a technique that systematically analyses complementary graphs of graphs under consideration.