SPECTRAL BOUNDS FOR LAPLACIAN ENERGY VIA WEIGHTED EIGENVALUE DEVIATIONS

Abstract

In this paper, we study Laplacian energy using a parameterized spectral framework based on deviations of Laplacian eigenvalues from the average degree. The approach introduces a weighting parameter that allows the contributions of individual eigenvalues to be examined more directly.
Using this formulation, we derive a partitioned lower bound and a quadratic spectral upper bound, both expressed in terms of Laplacian eigenvalues. The bounds depend on a continuous parameter and a discrete partition index, which are optimized according to the structure of the spectrum.
An asymptotic analysis is carried out for representative graph families, showing that the behaviour of the bounds depends on the distribution of the Laplacian eigenvalues. Graphs with concentrated or highly skewed spectra lead to limiting parameter values, while more balanced spectra lead to intermediate values.
Numerical results support these observations and show that the parameters reflect how the Laplacian energy is distributed across the spectrum. In this way, the framework also provides information about the underlying spectral structure of the graph, in addition to bounding the energy.