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

Dipakkumar N. Barad; Vinodray J. Kaneria; Kalpesh M. Popat

doi:https://doi.org/10.14445/22315373/IJMTT-V72I7P103

International Journal of Mathematics Trends and Technology

Research Article | Open Access | Download PDF

Volume 72 | Issue 7 | Year 2026 | Article Id. IJMTT-V72I7P103 | DOI : https://doi.org/10.14445/22315373/IJMTT-V72I7P103

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

Dipakkumar N. Barad, Vinodray J. Kaneria, Kalpesh M. Popat

Received	Revised	Accepted	Published
25 May 2026	30 Jun 2026	16 Jul 2026	30 Jul 2026

Citation :

Dipakkumar N. Barad, Vinodray J. Kaneria, Kalpesh M. Popat, "Maximum Degree Energy and Minimum Degree Energy in the Context of Some Graph Operations," International Journal of Mathematics Trends and Technology (IJMTT), vol. 72, no. 7, pp. 14-24, 2026. Crossref, https://doi.org/10.14445/22315373/IJMTT-V72I7P103

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.

Keywords

Extended 𝑚-Splitting, 𝑚-Semi Shadow, Maximum Degree Energy, Minimum Degree Energy, Regular Graphs.

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