Some new results on the Seidel energy of graphs with self-loops

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.

💡 Research Summary

This paper investigates the Seidel energy of graphs that may contain self‑loops, extending recent work by Harshitha et al. (2024). The authors adopt the definition of the Seidel matrix for a looped graph G_W (where W⊆V(G) denotes the set of vertices equipped with a self‑loop) as S(G_W)=S(G)−I_W, with I_W the diagonal matrix that has ones exactly on the entries corresponding to W. Consequently, the Seidel energy of a looped graph is SE(G_W)=∑_{i=1}^n|θ_i(G_W)+σ|, where σ=|W| and θ_i(G_W) are the eigenvalues of S(G_W). This formulation coincides with the matrix‑energy concept E(M)=∑ singular values of M, because for a real symmetric matrix the singular values are the absolute eigenvalues.

The first major contribution (Theorem 3.1) establishes sharp two‑sided bounds relating SE(G) and SE(G_W): \