Geometric And Algorithmic Aspects of Graph Representations: A Review of Spectral Methods, Graph Energy, and Structural Parameters
DOI:
https://doi.org/10.65204/djes.v3i3.927Keywords:
Extremal Graph Theory , Graph Energy, Laplacian Spectrum, Structural Graph, Algebraic ConnectivityAbstract
The two concepts of graph energy and Laplacian spectral theory function as essential components for contemporary algebraic and extremal graph research which provides essential tools for mathematical chemistry and complex network research. The article provides an extensive overview of recent progress about classical graph energy and Laplacian and singles Laplacian spectra and their complex connections with basic structural characteristics of graphs. The research focuses on presenting precise upper and lower limits together with their extreme characterizations and Nordhaus-Gaddum-type inequalities and spectral conditions, which depend on the connectivity and diameter and degree-based features of the network. The analysis expands into various advanced definitions of graph energy while showcasing the progress made in theoretical understanding during the past ten years. The survey concludes with a discussion of open problems and emerging research directions, aiming to provide a unified and critical perspective that may stimulate further investigations in spectral and structural graph theory.
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