Nested graphs with bounded second (signless Laplacian) eigenvalue

Nested split and double nested graphs (commonly named nested graphs) are considered. General statements regarding the signless Laplacian spectra are proven, and the nested graphs whose second largest signless Laplacian eigenvalue is bounded by a fixed integral constant are studied. Some sufficient conditions are provided and a procedure for classifying such graphs in particular cases is provided. Some connections between their structure and some (not only the second) eigenvalues of their signless Laplacians are developed. All double nested graphs whose second largest eigenvalue does not exceed √ 2 are determined.