The Laplacian Spectrum of a Graph

Abstract

Summary:In this paper we consider the energy of a simple graph with respect to its Laplacian eigenvalues, and prove some basic properties of this energy. In particular, we find the minimal value of this energy in the class of all connected graphs on $n$ vertices $(n=1,2,\ldots )$. Besides, we consider the class of all connected graphs whose Laplacian energy is uniformly bounded by a constant $\alpha \ge 4$, and completely describe this class in the case $\alpha =40$