Graph Energy of Non-Coprime Graphs for Some Finite Non-abelian Groups - Suleiman, Aliyu; Ibrahim Kiri, Aliyu; Tijjani, Bilkisu (Universitas Brawijaya, 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 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.