Maximum Degree Energy and Minimum Degree Energy in the Context of Some Graph Operations

Abstract

Let 𝐺 be a 𝑘-regular graph on 𝑛 vertices. In this paper, we determine the maximum and minimum degree energies of two specific graph operations: the extended 𝑚-splitting graph. 𝑆𝑝𝑙𝑚∗ (𝐺) and the 𝑚-semi shadow graph 𝑆𝐷𝑚(𝐺). By applying block matrix decompositions and unitary similarity transformations, we express the maximum and minimum degree spectra of these constructs explicitly in terms of the ordinary adjacency spectrum of the base graph 𝐺. Furthermore, we derive exact, closed-form expressions for their respective maximum and minimum degree energies. As applications of our main theorems, the corresponding energies for well-known families of regular graphs—including cycle, complete, and complete bipartite graphs—are explicitly established.