Graph Energy of Non-Coprime Graphs for Some Finite Non-abelian Groups - Suleiman, Aliyu; Ibrahim Kiri, Aliyu; Tijjani, Bilkisu (Indonesian Journal of Mathematics and Applications, 2026)
In this paper, we study the Laplacian energy and Seidel energy of the non-coprime graphs for the dihedral group $D_n$, $n ≥ 3$, $n = p^r$, where $p$ is prime and $r \in \mathbb{Z}^+$, and the generalized quaternion $Q_{4n}$, $n ≥ 2$. The investigation starts by obtaining the spectrum for the graphs and then generalizing the formulae for computing the respective energies. It is shown that the Seidel energy for the non-coprime graph for a dihedral group when $n = 2^r$, $r > 1$, is less than the Laplacian energy for the same graph; this is also true for the generalized quaternion. Furthermore, the Seidel energy for the non-coprime graph for a dihedral group $D_n$ when $n = 2^r$, $r > 1$, is the same as the adjacency energy of the same graph for the same group.