Generalized Graph Energies of a Regular Graph under Vertex Duplication Operation

Abstract

We provide a thorough examination of the graph energies in regular graphs that arise from the vertex duplication process in this paper. Understanding the numerous structural components of graphs requires understanding the thought of graph energy, known as a measurement obtained by computing eigenvalues the adjacency matrix of a graph. We derived generalized closed-form expressions for a number of important energy metrics, such as minimum degree, energy maximum degree energy, first Zagreb energy, second Zagreb energy and degree square sum energy, utilizing proficient algebraic graph theory techniques and eigenvalue spectrum analysis. Our work emphasizes on vertex duplication techniques and the impact they have on these energy metrics, primarily on regular graphs, a basic class of graphs where every vertex has the same degree. The resulting formulations offer further explanations for the behavior and attributes of these energy functions within the framework of regular graphs, providing a more comprehensive knowledge of how these operations affect the structural complexity of the graph. These findings greatly expand the conceptual model of graph energy and have potential uses in fields like combinatorics, chemistry, and network analysis where the energy models of graphs are extensively employed.

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