Independence number and the normalized Laplacian eigenvalue one

Abstract

In this paper, we prove that every simple connected graph with [Formula: see text] has [Formula: see text] as a normalized Laplacian eigenvalue with multiplicity at least [Formula: see text], where [Formula: see text] and [Formula: see text] are the order and the independence number of the graph, respectively. Then we investigate graphs with [Formula: see text] and having or not having [Formula: see text] as a normalized Laplacian eigenvalue.