Some new results on the Seidel energy of graphs with self-loops
Kalpesh M. Popat, Irena M. Jovanovic
TL;DR
This work advances the study of Seidel energy for graphs with self-loops by establishing a necessary and sufficient condition for when the Seidel energy of a looped graph $G_W$ equals that of its underlying graph $G$, showing this equality occurs only in the trivial loop cases $|W|=0$ or $|W|=n$. It derives tight bounds linking $\mathcal{SE}(G_W)$ to $\mathcal{SE}(G)$ via a perturbation argument and leverages Fan-type results to characterize equality. The paper also proves invariance of Seidel energy under complement and looped Seidel switching for looped graphs, and analyzes the Seidel energy of unions and joins: for a regular graph $G$ and its looped copy, the energy of $\mathcal{U}=G\cup G'$ satisfies $\mathcal{SE}(\mathcal{U}) = 2\mathcal{SE}(G) + 2(\max\{\lvert\theta_1(G)\rvert, R\} - \lvert\theta_1(G)\rvert)$ with $R=\sqrt{n^2+\tfrac{1}{4}}$, reducing to $2R+2\sum_{i=2}^n \lvert\theta_i(G)\rvert$ under a mild bound. These results extend Seidel-energy theory to looped graphs and provide explicit spectral formulas for composite constructions.
Abstract
Harshitha et al. recently introduced Seidel energy of graphs with self loops. In this paper, we extend some of their results by giving a necessary and sufficient condition for the Seidel energy of a looped graph to be equal to the Seidel energy of its underlying graph. We also consider Seidel energy of the union of certain graphs, and show that graph operations complement and Seidel switching preserve Seidel energy in the looped setting.